## Abstract

We evaluate the possibility to focus scanning light beams below the diffraction limit by using the combination of a nonlinear material with a Kerr-type nonlinearity or two-photon absorption to create seed evanescent components of the beam and a negative-refraction material to enhance them. Superfocusing to spots with a FWHM in the range of 0.2λ is theoretically predicted both in the context of the effective-medium theory and by the direct numerical solution of Maxwell equations for an inhomogeneous photonic crystal. The evolution of the transverse spectrum and the dependence of superfocusing on the parameters of the negative-refraction material are also studied. We show that the use of a Kerr-type nonlinear layer for the creation of seed evanescent components yields focused spots with a higher intensity compared with those obtained by the application of a saturable absorber.

© 2006 Optical Society of America

## 1. Introduction

In many fields and applications such as optical lithography[1], optical data storage[2], nanoprocessing and others, the smallest spot size of a focused optical beam limits downsizing of optical devices or marks towards the nanometer scale. In conventional optics, diffraction limits the FWHM of the smallest spot to half of the wavelength, because all Fourier components with transverse wavenumber larger than 2*π*/λ (evanescent components) are lost during propagation. Near-field optical methods can overcome this limitation due to the creation and interaction of evanescent components with the sample. One of the most common ways is the use tapered or etched optical fibers [3] that confine a light spot to a nanometer area. Solid immersion lenses[4] can also produce subdiffraction spots by the application of high-refractive-index solids as lens material in the near-field range of the sample. However, the main problem for the application of these techniques is the required subwavelength proximity of the moving nearfield element and the medium which is to be studied or modified. Although impressive results has been achieved in this field[5], the required spatial near-field control, low scanning speed and reliability, and small throughput remain formidable obstacles for applications.

The aim of this paper is to study an alternative method for focusing of scanning beams below the diffraction limit which relies on the application of two main elements: A nonlinear layer which creates evanescent components from an optical beam and a layer with a negative index (superlens) which amplifies the evanescent components and focus the beam below λ/2.

Negative refraction materials [6] (NRM) have recently attracted great interest because of their fascinating novel properties, in particular the possibility to amplify evanescent components of light for subwavelength imaging [7]. These findings stimulated intense experimental and theoretical research in this field. Up to now NRM has been realized mainly in two types of systems: metamaterials which combine electric and magnetic resonances to obtain both *ε* and *μ* below zero[8], and photonic crystals with negative refraction in certain frequency ranges[9, 10]. Within the first type of materials, negative refraction has been studied for various systems such as split ring resonators[8], parallel nanowires and nanoplates [11, 12] and coupled nano-cones [13]. In addition, there even exist a theoretical predictions of negative refraction in completely homogeneous materials such as dense noble gases[14] using electromagnetically induced transparency. Besides, amplification of evanescent components can also be achieved in photonic crystals with negative refraction in parameter regions without a definite refraction index due to coupling to bound photon modes (so-called all-angle negative refraction)[15]. Even a simple metal slab can amplify evanescent components and act like a superlens[16, 17]. These effects are demonstrated for wavelength ranges extending into the NIR and visible[11, 12, 13, 18, 19] and for both 2D and 3D[20] structures. Based on these findings, up to now most effort for applications of NRMs were related to the use of the superlensing effect for imaging of subwavelength features of point-like sources positioned in the near-field range of the NRM lens[15, 21, 22]. In this case, to achieve subdiffraction resolution, one just has to recover the amplitude of the evanescent components which are already present at the input. The presence of both propagating and evanescent components results in an image with subdiffraction resolution, with spectral phases being compensated due to the geometry of the flat lens.

Recently the application of NRM was also proposed for focusing of optical beams below the diffraction limit[23]. Since in an input beam no evanescent components are present which could be amplified by a NRM, an additional element, like an aperture[23], placed directly before the superlens is necessary to create seed evanescent components from the beam. For focusing of scanning beams a fixed position of an aperture do not allow to focus the beam at an arbitrary point. Therefore in Ref. [24] a saturable absorber was proposed as a light-controlled nonlinear element creating evanescent components from the beam. However, to achieve a sufficient magnitude of seed evanescent components, strong absorption was found to be necessary, leading to a relatively large loss with an intensity in the focus at the level of 10^{-2}–10^{-1} of the input intensity.

In this paper, we theoretically study subdiffraction focusing of scanning light beams by the combination of a Kerr-like nonlinear layer and a layer of NRM, as illustrated in Fig. 1(a). In Fig. 1(b), the creation of the evanescent components *E*_{evan}
by a Kerr medium (KM) and its amplification by a NRM is schematically shown. The phases of both propagating and evanescent components are almost matched for optimized parameters, which allows constructive interference of all components and the formation of a subwavelength spot. Subdiffraction focusing is described both in the context of the effective-medium description for quasi-homogeneous NRMs, such as metamaterials, and by direct numerical solution of Maxwell equations in a periodic medium for a NRM implemented by a photonic crystal. We show that in order to avoid loss of the evanescent components and diffraction of the input beam it is necessary to choose a nonlinear material with a high linear refractive index. Additionally, we investigate the dependence of subdiffraction focusing on the deviations of the NRM parameters from the ideal *ε*=*μ*=-1 case and determine the parameters which most sensitively influence the focusing. We have also studied a system with a nonlinearity provided by a two-photon absorber instead of a Kerr medium, which allows to lower intensity requirements. We note that in contrast to results of Ref. [24] the high transmission of propagating components in the Kerr-like nonlinear medium allows to achieve focusing with intensity close to or higher than the input intensity.

In Section 2 the superfocusing is described by the effective-medium theory, while in Section 3 results are presented from the numerical solution of Maxwell equations for photonic crystals as NRM. The superfocusing with two-photon absorber as a nonlinear element is considered in Section 4, and Section 5 contains the conclusion.

## 2. The effective-medium-theory description of NRM

The modification of the refractive index by intense light is one of the fundamental effects in nonlinear optics. In many cases such modification is achieved due to the electronic Kerr effect, but there exist a number of other processes which can also lead to a high nonlinear refractive index. Although those processes with the highest nonlinear coefficients have typically a large response time, ultrafast operation is not a critical requirement for the purpose studied here. For the description of beam propagation through a nonlinear layer including the generation of evanescent components the paraxial approximation cannot be used. An exact treatment of this process is based on the general wave equation written for a monochromatic field in the form

where
${k}_{z}=\sqrt{{n}_{0}^{2}{k}_{0}^{2}-{k}_{x}^{2}-{k}_{y}^{2}},{k}_{0}=\frac{{\omega}_{0}}{c},{n}_{0}$
is the refractive index of the nonlinear layer, *ω*
_{0} is the frequency of the field, *P*_{NL}
(*k*_{x}
, *k*_{y}
, *z*) denotes the Fourier transform of the nonlinear polarization *P*_{NL}
(*x*, *y*, *z*)=*ε*
_{0}
*χ*
_{3}
*E*(*x*, *y*, *z*)^{3}, *χ*
_{3}=(4/3)*c*ε_{0}
${n}_{0}^{2}$
*n*
_{2} is the nonlinear susceptibility. In the perturbation-theory approach the solution can be found in the form

$$\frac{\mathrm{exp}\left(i{n}_{0}{k}_{0}z\right)-\mathrm{exp}\left(-i{k}_{z}z+i{k}_{z}L+i{n}_{0}{k}_{0}L\right)}{{n}_{0}{k}_{0}+{k}_{z}}]$$

for *z*<*L* where *L* is the thickness of the nonlinear layer. The first term in the square brackets corresponds to the forward-propagating waves, and the second to the backward-propagating waves which is zero for *z*>*L*. It can be shown that the forward-propagating part satisfies a simpler first-order equation:

Inside the nonlinear layer the nonlinear polarization is also influenced by the weak backward waves, but this effect is negligible for thin layers with a small relative modification of the refractive index. Equation (3) is valid also without the perturbation theory, and can be derived using quite general assumptions as shown in Ref. [25]. This equation does not rely on the paraxial approximation and, as an additional feature, correctly describes the evolution of the components with ${k}_{x}^{2}$+${k}_{y}^{2}$>${n}_{0}^{2}$
${k}_{0}^{2}$. Note that the nonlinear term has a divergence at *k*_{z}
=0 in the Fourier domain, but after Fourier transform to the space the field *E*(*x*, *y*, *z*) remains finite. Besides, Fourier components with *k*_{z}
=0 are very weak and do not contribute to the formation of the spot in our calculation. Equation (3) was solved by the split-step Fourier method.

For a homogeneous or quasi-homogeneous medium with negative refraction index the effective-medium theory can be used to describe the propagation of the beam and the amplification of evanescent components in a NRM. Various negative-refraction metamaterials based on structures with a scale much smaller than the wavelength can be described by this method. In the context of effective medium theory, the propagation of a beam through the homogeneous slab of the NRM with thickness *L* and effective parameters *ε* and *μ* is described by the transfer functions *T*_{s,p}
(*k*_{x}
, *k*_{y}
)

where κ_{s}=*μk*_{z}
/*q*_{z}
+*q*_{z}
/(*μk*_{z}
) for S-polarized and κ_{p}=*εk*_{z}
/*q*_{z}
+*q*_{z}
/(*εk*_{z}
) for P-polarized components with
${q}_{z}=\sqrt{\epsilon \mu {k}_{0}^{2}-{k}_{\perp}^{2}}$
,
${k}_{z}=\sqrt{{k}_{0}^{2}-{k}_{\perp}^{2}}$
,
${k}_{0}=\frac{2\pi}{\lambda}$
,
${k}_{\perp}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}$
, *s* and *p* denote correspondingly *S-* and *P-* polarization. These transfer functions relate the output field **E**
_{s,p}
(*k*_{x}
, *k*_{y}
, *L*) with the input field **E**
_{s,p}(*k*_{x}
, *k*_{y}
,0). The spatial structure of the field at the output of the system is then found by the backward Fourier transformation. The back-reflection from the Kerr medium is neglected. The field **E**
_{s,p}(*k*_{x}
, *k*_{y}
, 0) at the input surface of the NRM is given by the field after the slab of a nonlinear Kerr-type medium positioned immediately before the slab, as shown in Fig. 1.

In Fig. 2 the superfocusing of a light beam by the combination of a Kerr-type nonlinear layer and a NRM is illustrated, with parameters given in the caption. We describe the profile of the field at the input surface of the nonlinear layer as modified Gaussian with evanescent components set to zero. In Fig. 2(a) the spatial transverse Fourier distribution *E*(*k*_{x}
, *k*_{y}
) after the nonlinear layer is shown by the red surface, and weak evanescent components at a level around 10^{-3} relative to the maximum can be seen for *k*
_{⊥}~2*π*/λ. The transverse Fourier distribution after the slab of the NRM is shown in green, with clearly visible large amplification of evanescent components. The corresponding transverse spatial distribution is presented in Fig. 2(b). The minimum beam diameter is 0.22λ/0.15λ in the *x*/*y* direction, respectively.

Note that for a nonlinear layer with a refractive index *n*
_{0} around unity, the evanescent components are lost over the distance λ/2*π*, and therefore the effective length over which they are generated is of the same order. That means that to achieve significant (of about *π*/2) phase modification necessary to generate seed evanescent components of sufficient intensity, over such a short distance, the maximum nonlinear contribution to the refractive index should be around unity. Since materials with such high nonlinear modification of the refractive index are not available, we cannot use a nonlinear film with *n*
_{0}~1 and thickness around λ/2*π* as the nonlinear layer. Instead, we considered in Fig. 2 a nonlinear layer with relatively high linear refractive index *n*
_{0}=3.0 and thickness of about λ, which allows to preserve and accumulate the evanescent components with
${k}_{0}<\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}<{n}_{0}{k}_{0}$
. For clarity, in this paper the term “evanescent components” denotes all transverse Fourier components with
$\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}>{k}_{0}$
where *k*
_{0} is the *vacuum* wavenumber. In this case the effective length of the nonlinear layer is limited only by the gradual increase of the spot size due to diffraction, which occurs after a propagation distance of several λ, and a relative nonlinear modification of the refractive index in the order of 0.03 is sufficient. There exist a large number of natural and artificial materials with strong optical nonlinearities. To give a few examples, the nonlinear refractive index of different semiconductors can have *n*
_{2} values in the order of 10^{-12} W/cm^{2}. Due to near-resonant processes in the vicinity of the bandgap, the values in the range of 10^{-8} W/cm^{2} can be achieved, for example in ZnSe in the wavelength range from 440 to 460 nm[26]. Metal-dielectric multilayer structures can yield a complex *n*
_{2} in the order of |*n*
_{2}|=7×10^{-8} W/cm^{2}[27]. Extremely high nonlinearities *n*
_{2}~10^{0} -10^{3} W/cm^{2} can be achieved in thin dye-doped liquid-crystal layers[28]. One has to take into account that for slow processes with a large response time *T*_{res}
the effective nonlinear refractive index is given by *n*
_{2}
*τ*/*t*_{res}
where *τ* is the pulse duration. Depending on the chosen nonlinear material, the parameter Δ${n}_{\text{Kerr}}^{\text{max}}$/*n*
_{0}=*n*
_{2}
*I*/*n*
_{0}=0.03 which is assumed in Fig. 2 can be achieved with intensities in the order of 10^{7}–10^{12} W/cm^{2}.

Figure 3 characterizes the evolution of the transverse spectrum and the spatial distribution of the intensity in more detail. At *z*~1λ, one can see that evanescent components start to appear, and after the nonlinear layer they possess a notable amplitude. Note also that during the propagation in the NRM layer the optimum spot is achieved not at the position when the spectrum is broadest (*z*=2.4λ), but at the position *z*=2.6λ where the spectrum is flat and the phases (not shown) imply constructive interference.With further propagation, the spectrum becomes narrower due to the assumed deviations of *ε* and *μ* from the ideal case *ε*=*μ*=-1.

The nonlinear modification of the refractive index which was used in the above example corresponds to rather high value of Δ${n}_{\text{Kerr}}^{\text{max}}$/*n*
_{0}=3.3%. It is, fortunately, not an absolute prerequisite of superfocusing, and significantly lower values can be used, albeit with tradeoff of the spot size. In Fig. 4 the superfocusing is illustrated for a one order of magnitude lower nonlinear modification of the refractive index Δ${n}_{\text{Kerr}}^{\text{max}}$/*n*
_{0}=3×10^{-3}, which can be achieved even by a fast Kerr nonlinearity. The transverse spectrum is narrower than in Fig. 2 due to a lower amplitude of the seed evanescent components, and the output spot is larger with a FWHM of 0.38λ/0.25λ, but still exhibits focusing below the diffraction limit.

The role of the deviations *δε*=*ε*+1 and *δμ*=*μ*+1 from the ideal -1 values is described in Fig. 5, where the FWHM spot area is plotted against the level of different deviations, as indicated in the caption. It can be seen that the most sensitive parameter is the imaginary part of *ε* and *μ* (red solid curve) corresponding to loss, which thwarts superfocusing even faster than the simultaneous deviations of all parameters from the ideal case (black dotted curve). The reason is that although the spectrum is broader in the case when only loss is present, the phases of the additional spectral components can lead to destructive interference, yielding a larger spot. The dependence of the superfocusing on the Re(*δε*) is – somewhat surprisingly – the least sensitive. As can be seen, a very low level of deviations is required to achieve superfocusing, similar to subdiffraction imaging[29], however, the requirements are much less stringent in the case of photonic crystals considered in the next section.

## 3. Superfocusing by a nonlinear layer and a photonic crystal with negative refraction

The effective-medium description is a sufficient approximation for a NRM based on metamaterials, which have structures much smaller than the wavelength. However, for photonic crystals with a wavelength-scale structure the assumption of quasi-homogeneity is not valid, and the effective medium theory yields only a qualitative description for such systems. Photonic crystals constitute an important possible realization of materials with negative refraction, which has been extensively studied recently, both experimentally and theoretically. To quantitatively describe superfocusing by the combination of a nonlinear layer and a photonic crystals in an inhomogeneous negative-refraction system, we now study a 2D photonic crystal with geometric parameters appropriate for an experimental realization.

For a photonic crystal the solution of Maxwell equation is found by the decomposition of the field into a large (~10^{7}) number of planar waves, including the reflected and transmitted plane waves outside the slab, as detailed in Ref. [30]. The periodic boundaries of the structure determine the coupling between these waves, resulting in a large-order system of linear equations which is solved numerically. We consider the parameter region with all-angle negative refraction of a hexagonal lattice of circular holes in the material like Si with parameters and geometry as indicated in Fig. 6(a). In Fig. 6(b), the transmission into the 0th Bragg order is presented in dependence on the transverse wavenumber of the incoming wave, which shows several peaks for evanescent components with *k*
_{⊥}>2*π*/λ with a transmission up to 10^{3}.

In Figure 7, superfocusing by the combination of a nonlinear layer and a photonic crystal slab is illustrated by the calculated transverse spectrum (a) and the spatial distribution (b), with parameters of the photonic crystal as given in Fig. 6. The amplification of the evanescent components leads to the formation of a wide transverse spectrum with a phase which varies mostly in the range from 0 to π/3 over the whole range where the spectrum has significant magnitudes. This implies the formation of a localized spot in space, and indeed as presented in Fig. 7(b) the FWHM of the spot is 0.18λ, significantly below the diffraction limit. The seed evanescent components are created mostly at the center of the input beam and in phase with the input beam for a Kerr-like nonlinear layer. Therefore the phase-matching at the output depends mainly on the phase of the transmission coefficient, which can be made almost constant by optimizing the parameters of the system. The peak intensity is approximately two times higher that the input intensity. In comparison, superfocusing by the combination of a saturable absorber and a photonic crystal with negative refraction yields a relative intensity of the focused spot in the range of 10^{-2}–10^{-1}, and somewhat larger spot sizes [24]. On the other hand, the required intensities for superfocusing are in that case lower, and the transmission in Ref. [24] is still significantly higher than that obtained using a tapered fiber as near-field element.

## 4. Superfocusing with nonlinearity provided by a two-photon absorber

Additionally, we study two-photon absorbers as alternative way to achieve the generation of seed evanescent components. An intensity-dependent absorption coefficient *α*(*I*)=*βI* leads to a nonlinear transmission *T*(*I*)=1/(1+*α*
_{0}
*Iz*) which provides weak evanescent components. The general mechanism of superfocusing is similar to the process described above. In Fig. 8, superfocusing is predicted both for a NRM described by the effective-medium theory and for a photonic crystal, with output spots with FWHM 0.32λ/0.2λ and 0.31λ, correspondingly, for parameters given in the caption. Note that the optimum photonic crystal in this case is different from that studied in Section 3, but the design used in this Section was found optimum for the case when seed evanescent components were created by an aperture[23]. The reason is that, unlike in the case of a Kerr-type nonlinearity, both an aperture and a two-photon absorber tend to change the input beam toward a top-hat transverse shape. Therefore evanescent components have a similar dependence of the phase on the transverse wavenumber in the cases of an aperture and a saturable absorber. The required nonlinear contribution to the transmission can be achieved with intensities in the order of 10^{7} W/cm^{2} in materials with giant two-photon absorption, like CuCl, characterized by a two-photon absorption coefficient of 10^{6} cm/GW[32], for a thickness of the absorber layer in the order of λ. Although a larger spot with stronger side maxima are achieved in this case in comparison with the case of Kerr-type nonlinearity, the required intensity is smaller.

## 5. Conclusion

We have predicted that superfocusing of scanning beams below the diffraction limit is possible using the combination of a nonlinear layer with a Kerr-like nonlinearity which generates seed evanescent components, and a layer with negative refraction which amplifies them. This method do not require spatial near-field control of a moving near-field element and allow an arbitrary position of the focused spot of scanning beams due to the action of the light-controlled nonlinear layer. The creation and amplification of evanescent components by these two elements allow the formation of a 0.18λ -wide focused spot, with matched phases of the transverse spectral components implying constructive interference, and an intensity two times larger than the input intensity. Superfocusing is demonstrated both in the effective-medium theory for a metamaterial-based NMR and for photonic crystals with realistic parameters. In comparison with an arrangement using the combination of a saturable absorber for the creation of seed evanescent components and a NRM, the focused spot intensity is more than one order of magnitude higher, but the required input intensity is also higher. Alternatively, giant two-photon absorbers do not require such high input intensity and can also provide subdiffraction spots.

Although the results of our study using the effective medium theory, and the numerical calculation for an photonic crystal predict similar results concerning the spot size and the optimum geometrical parameters, there exist an important difference: the high sensitivity of superfocusing on the deviations of *ε* and *μ* from -1 in the effective medium theory does not arise for photonic crystals and superfocusing is even possible in the range of so-called all-angle negative refraction where an effective index can not be defined. The physical mechanism here is that wide transmission peaks for evanescent components due to resonances with bound photon modes yield a sufficient amplification of the evanescent components which allow superfocusing. This means that not the negative index is the main physical requirement for the predicted effect, but the existence of wide transmission peaks much larger than unity for evanescent components.

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