Focusing of light beams into an extremely small spot with a high energy density is an important issue in optics. High-numerical-aperture focusing is used as well in key technologies for miniaturization like lithography, optical date storage, in laser material nanoprocessing and nanooptics as in confocal microscopy and numerous other fields. Pushing the spot size below the diffraction limit is of exceptional importance for many of these applications. The physical limit for the smallest spot size of focused light for all conventional optical devices is determined by the fact that all evanescent waves of an electromagnetic wave are lost during propagation. The reason is that all the components E(k_x ,k_y ,z) in the spatial Fourier presentation of the electric field E(x,y,z,t)=exp(-iωt)∫∫E(k_x ,k_y ,z)exp(ik_xx+ik_yy)dk_xdk_y with transverse wavenumber k _⊥=($k_x^2$ +$k_y^2$ )^1/2 which satisfy the relation k _⊥>2π/λ (so-called evanescent waves) decay exponentially as exp(ik_zz) because for these components the longitudinal wavenumber k_z =($k_0^2$-$k_{\perp }^2$)^1/2 becomes imaginary. Here λ=2πc/ω is the vacuum wavelength, k ₀=2π/λ=ω/c and z the propagation length.

The diffraction limit can be circumvented by using near-field optical techniques such as tapered fibers with a metalized thin tip placed very close to the object or a solid immersion lens[1]. In the present Letter an alternative method to achieve superfocusing is studied based on the suggestion of Pendry[2] that evanescent waves can be amplified by a slab with a negative refraction index leading to a superlensing effect with a sub-wavelength resolution image of a point source. The electrodynamics of materials with negative refractive index was first studied several decades ago [3] and later such phenomenon was experimentally demonstrated for microwaves in a meta-material using a combination of copper rings and wires [4]. Recent theoretical and experimental work [5, 6, 7, 8] has shown that an effective negative refraction index in the near-infrared and optical range can also be realized using strongly modulated photonic crystals (PC) below the bandgap edge. Moreover, the amplification of evanescent waves can be achieved in these materials even in parameter regions without a negative refraction index due to resonant coupling to bound photon states [7, 9, 10]. Up to now the superlensing effect was studied for imaging of sub-wavelength features [11, 7, 10, 12]. Here we explore the possibility to amplify the evanescence components for the aim of sub-wavelength focusing of an electromagnetic beam below the diffraction limit. Although both physical issues are closely related, there exist crucial differences: in contrast to imaging, in the case of focusing no seed evanescent waves are present in the input beam which could be amplified by the PC. Therefore we introduce directly before the slab of the PC a wavelength-scale aperture which creates from the input beam seed evanescent components of sufficiently large amplitude. These evanescent fields are amplified by the PC with a thickness L in the range of λ. We show that using an aperture with a diameter in the range of 1–2 λ and a PC with negative refraction, a focused spot with a FWHM of about ~0.25λ can be achieved. This spot cannot be an image of the much larger aperture, but results from the wide spatial spectrum obtained by amplification of the seed evanescent components. Furthermore, we show that the constructive superposition of these amplified evanescent components can be achieved.

To get insight into this phenomenon we first describe beam focusing by a plane slab of a medium with a negative refraction index in the context of effective medium theory. The propagation of a beam through the slab can be described, similar to a positive refraction layer, solving the Maxwell equation for the spatial Fourier components and using the boundary conditions on both sides of the slab. Then for arbitrary ε and µ the output field is related with the input field by E _s,p(k_x ,k_y ,L)=T_s,p (k_x ,k_y )E _s,p (k_x ,k_y ,0), where T_s,p (k_x ,k_y ) are the transfer functions which are diagonal in the Fourier presentation due to the spatial homogeneity of the slab

Here κ=µk_z /q_z +q_z /(µk_z ) for S-polarized and κ=εk_z /q_z +q_z /(ε k_z ) for P-polarized components with q_z =(${\epsilon \mathit{\mu }k}_0^2$-$k_{\perp }^2$)^1/2. We neglect the contribution from reflection from the slab followed by back-reflection from the aperture. The spatial structure of the field at the output of the system is then found by the backward Fourier transformation. The input field E _s,p (k_x ,k_y ,0) here is given by the field after the circular aperture in a perfectly-conducting infinitely-thin film positioned immediately before the slab. In the case of realistic (for example metallic) film of finite thickness, the diffraction will be more complicated due to various effects such as excitation of surface plasmons. However, for the realistic film parameters and wavelength in near IR or visible, coupling to the plasmons will be efficient only near the aperture. Thus, they will not significatly influence the field distribution after the film. The solution of the diffraction problem of a beam through a circular aperture with a diameter in the range of the wavelength [13], provides the diffracted field after the aperture E(k_x ,k_y ,0). The crucial point here is that in difference to the incoming beam before the aperture, E(k_x ,k_y ,0) contains propagating components as well as evanescent components, the latter of which can be amplified by the slab with the negative refraction index.

 

Fig. 1. Superfocusing in the effective-medium theory with a negative refractive index. The spatial spectrum |E(k_x ,k_y )| is shown in (a) after the aperture |E(k_x ,k_y ,0)| (red, surface I) and after the slab |E(k_x ,k_y ,L)| (green, surface II). The spatial distribution of the energy density after the slab is shown in (b) and the dependence of the FWHM diameters along x (triangles) and y (circles) directions in (c). The parameters for (a),(b) are L=0.4λ and ε=µ=-1-0.0001(i+1). In (c), ε=µ=-1-0.0001(i+1) and ε=µ=-1-0.01(i+1) for solid (green) and open (magenta) points, correspondingly. Curves in (c) are guide for the eyes; the aperture diameter is 2λ.

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In Fig. 1(a) the surface I (red) represents the spatial transverse Fourier distribution after the aperture for a x-polarized input beam with parameters as given in the caption, where weak evanescent components at the level 10^-2 can be seen for k _⊥~2.0k ₀. During the transmission through the slab the evanescent components are strongly amplified. Though the thickness of the slab in this case is only 0.5λ, the amplification of the field for k _⊥=3k ₀ is around 10⁴, which explains the appearance of the ‘ring’ of the amplified components around k _⊥=3.5k ₀. The corresponding transverse spatial distribution of the energy density including the longitudinal component with a strongly focused peak (superfocusing) is illustrated by Fig. 1(b). The FWHM of this peak is 0.15λ/0.12λ in the x/y direction, respectively; the difference arises due to difference in diffraction and transmission for the different vector components of the field. The dependence of the output FWHM on the slab thickness is presented in Fig. 1(c) by solid (green) and open (magenta) points for ε=µ=-1-0.0001(i+1) and ε=µ=-1-0.01(i+1), respectively. The deviations of ε and µ from the optimum value -1 limit the superfocusing (cf. Ref. [14]). The existence of an optimum slab thickness [minima of curves in Fig. 1(c)] is caused by the fact that the increasing amplification of evanescent components with larger slab thickness is counteracted by the decrease of the transmission coefficients due to the deviations from ideal value of ε=µ=-1. For the higher level of deviations 10^-2, the optimum FWHM at the output are 0.23λ/0.33λ, which is still below the diffraction limit.

 

Fig. 2. Superfocusing in a hexagonal rods-in-air lattice PC with an effective negative refraction index. Transfer into the 0th Bragg order is shown in (a), the spatial spectrum (solid, red) and phase (dashed, green) of the field H_z are given in (b). The spatial distribution of the field is shown in (c), and the dependence of the FWHM of the beam at the output (solid, red) and the square of the maximum field |H_z (0)|² (dashed, green) on the aperture width in given in (d). The input beam has a Gaussian shape with FWHM of 3.2λ, the photonic crystal slab consists of 4 layers of circular rods with radius 0.35a with lattice constant a. The frequency of the field is 0.57×2πc/a, the dielectric permittivity of the rods ε=12.96+0.01i. The aperture width in cases (b) and (c) is 0.55λ.

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To explore the phenomenon of beam superfocusing in a real physical system, we now study this effect in one transverse direction by a two-dimensional (2D) photonic crystal. For certain design parameters and below the bandgap edge strongly modulated photonic crystals behave as a material having an effective negative refraction index [5]. Evanescent components are amplified in such photonic crystals analogous to the effective medium model. Besides it was shown that the amplification of evanescent waves in PCs can also be achieved for regions with an effective positive refraction index due to resonant coupling with bound photon modes, which has been denoted as all-angle negative refraction [7]. Here we will consider both superfocusing in a PC with an effective negative refractive index as well as in regions with all-angle negative refraction. The 1D aperture is considered infinitely thin and ideally-absorbing in this case. The propagation of the TE-polarized beam through the 2D PC is calculated by the corrected plane-wave expansion method (see e.g. Ref. [15]), with Bragg orders up to 20 included into the calculation. Due to the significant contrast of the dielectric permittivity, a large number of the plane-wave coefficients (~10⁴) has been used. First we consider a rods-in-air structure with a periodic arrangement of silicon rods (with ε=12+0.01i) with diameter 0.7a where a is the lattice constant. This PC possess a negative refractive index for frequencies around ω=0.56×2π c/a [10]. The transfer function into the forward direction (0th Bragg order) is presented in Fig. 2(a), and the amplitude of the Fourier components after the slab is presented by the solid (red) curve in Fig. 2(b). As can be seen, evanescent components with k_x >k ₀ are strongly amplified, and play a crucial role in the formation of the focused spot. The phase of the Fourier-transformed field [dashed (green) curve in Fig. 2(b)] shows large changes in the whole presented k-range, however it only weakly varies inside the range with a large Fourier amplitude. Therefore the superposition of all Fourier components is mainly constructive. The spatial transverse field structure is presented in Fig. 2(c) by the distribution of the |H_z |² component of the field (relative to the input field). The field at the output is weaker than at the input, because of the reflection at the slab interfaces caused by the refractive index contrast. The FWHM is 0.25λ in this case, however there exist a modulated broader background, and in Fig. 2(d) it is shown that the range of the aperture diameters for superfocusing is very narrow.

 

Fig. 3. Superfocusing in a square air-holes lattice PC with an all-angle negative refraction. Transfer into the 0th Bragg order is shown in (a), the spatial spectrum (solid, red) and phase (dashed, green) of the field H_z is given in (b). The spatial distribution of the field is shown in (c), and the dependence of the FWHM of the beam at the output (solid, red) and the square of the maximum field |H_z (0)|² (dashed, green) on the aperture width is given in (d). The input beam has a Gaussian shape with FWHM of 3.2λ, the photonic crystal slab consists of 4 layers of ‘+’-shaped air holes with parameters as given in Fig. 4(b). The frequency of the field is 0.27×2πc/a, the dielectric permittivity of the bulk is ε=12+0.01i. The aperture width in cases (b) and (c) is 1.15λ.

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Now we study beam focusing by a 2D photonic crystal with an effective positive refraction index but within the range of so-called all-angle negative refraction which arises due to resonances with bound photon modes. We consider a photonic crystal made of ‘+’-shaped air holes in a high-index material. Such a PC made from Silicon (ε=12) with geometric parameters as given in Fig. 4(b) exhibits all-angle negative refraction near the frequency 0.27×2π c/a [7]. As can be seen in Fig. 3(a), the evanescent components of the field are also amplified in this system. In difference to the case presented in Fig. 2 now sharp peaks in the transfer function appear. These peaks influence the spatial Fourier distribution of the field at the output as shown in Fig. 3(b). The phase presented by the dashed (green) line in Fig. 3(b), shows a rather irregular behavior, however the phases are nearly matched in the dominating regions with large spectral amplitudes. Therefore the spatial distribution as shown in Fig. 3(c) shows a sharply focused spot with FWHM of 0.25λ, and only weak background radiation. The peak of |H_z (0)|² is approximately 4 times higher than at the input. The dependence of beam FWHM at the output of the PC on the aperture width is illustrated by Fig. 3(d) and shows a well-established minimum, a notable enhancement of the field, and superfocusing over a large parameter range.

It was shown in the previous studies (see e.g. [5] and [14]), that efficient amplification of the evanescent components is possible only in a narrow region of frequencies. We have observed similar effect: even a small change of the optical frequency by ±0.5% results in approximately a twofold increase of the FWHM and the disapearance of superfocusing.

 

Fig. 4. Distribution of the field of the beam (a) and the structure of the unit cell of the photonic crystal for the same system as in Fig. 3. The value of |H_z | is illustrated by shadows of blue in (a), and the aperture is represented by green; the reflected beam is omitted for the sake of clarity.

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To illustrate the process of superfocusing in more detail, we have calculated the spatial structure of the field inside and behind the PC as presented in Fig. 4(a). The input beam after passing the aperture (indicated by green) undergoes complex redistribution in the photonic crystal, and is strongly focused below the diffraction limit directly behind the PC. With further propagation, the FWHM of the beam quickly increases. We see that the considered PC with all-angle negative refraction is better suited for superfocusing than the PC with negative refractive index studied in Fig. 2.

In summary, we have shown that it is possible to achieve focusing of a light beam below the diffraction limit by using a combination of an aperture and a slab with a medium exhibiting negative refraction caused by the creation and the amplification of evanescent components of the field. Note that the fairly restricted region for the parameters µ and ε in the effective medium model is eliminated in real photonic crystals in parameter regions where the concept of effective negative-index is valid and in the region of all-angle negative refraction. Although the above analysis was done for a 2D photonic crystal, a 3D photonic crystals with all-angle negative refraction has been already designed and studied in Ref. [16] which allows superfocusing of light beams in both transverse directions similar to Fig. 1. The smallest spot size as found here can be further decreased using optimized PC design (in particular smaller a), optimized aperture forms as e.g. annular or quadrupole shapes and other improvements.