1. Introduction

Development of new mechanisms to control diffraction of focused electromagnetic radiation is a central problem in Optics, with implications in a wealth of application areas. Engineered artificial materials with complex refractive index landscapes [1,2] and metamaterials [3] provide wide potential opportunities for diffraction control due to the possibilities afforded by the corresponding spatial dispersion relations that characterize light propagation in the material. Diffraction management is especially interesting at the subwavelength scales, because of its potential applications for signal routing in compact devices [4], subwavelength imaging [5–7], or nanolithography [8,9].

In many schemes involving optical media with a linear response, diffraction management is addressed in structures whose dielectric permittivity varies rapidly along the direction of beam propagation. In some cases, such a rapid permittivity variation allows to use simple effective medium models to describe light propagation, assuming that radiation propagates in an effective homogeneous medium with the averaged dielectric permittivity. Diffractionless propagation of subwavelength beams were reported in inhomogeneous nanostructures designed in such a way, when the spatial dispersion relation yields nearly-flat equal-frequency surfaces for a wide range of spatial frequencies [10–14]. Diffractionless propagation has also been predicted in metal-dielectric stacks with hyperbolic dispersion [15–21], where the effective permittivity can be designed to have opposite signs for different polarizations [22]. However, when the amplitude of modulation of the dielectric permittivity becomes large in comparison with the mean value, a new regime emerges in which the beam evolution is affected not only by the value of the averaged permittivity, but also by additional important contributions arising from the rapidly-varying periodic permittivity oscillations. The appearance and implications of such terms are known as the Kapitza effect, and is a phenomenon that is encountered in several areas of classical and quantum physics. The effect has important implications to light propagation. For example, localized waveguides whose depth is strongly and periodically modulated in the longitudinal direction, have been recently shown to yield light confinement even when the average refractive index contrast vanishes [23]. Importantly, Rizza and Ciattoni predicted earlier in the microwave domain $(\lambda \sim 100\mu m)$that nearly diffractionless propagation is possible in Kapitza structures, even down to subwavelength scales [24]. Diffraction arrest was shown to occur because the rapid, large-amplitude permittivity modulations strongly suppress the longitudinal component of the electric field, thus slowing down diffraction of the transverse magnetic (TM) waves [24–26]. The effect was shown to occur with Gaussian-like beams. However, the ability of the Kapitza effect to suppress diffraction may vary depending on the spatial complexity of the beam, hence on the width of the corresponding spatial spectrum.

2. Model and discussions

In this paper we study the efficiency for diffraction cancellation of the Kapitza effect in the case of structured subwavelength light beams featuring closely-packed multiple-peak intensity distributions. Specifically, we study how the diffraction-cancellation efficiency of the Kapitza scheme decreases as the spatial spectral bandwidth of the input subwavelength beams grows. We also study the impact of the Kapitza effect on the propagation angle of titled input beams and its implications for beam steering of subwavelength light.

Without loss of generality, Kapitza medium here is characterized by smooth periodic variation of the relative dielectric permittivity along the $z$-axis:

where ${\epsilon }_{bg}$is the relative mean or “background” permittivity; the parameters $p$ and $\zeta$characterize, respectively, the depth and period of the permittivity modulation in the longitudinal $z$direction. The wavelength of laser radiation $\lambda =632.8\text{nm}$ is chosen from the visible range, while modulation period$\zeta$ should be notably smaller than the wavelength $\lambda$ to guarantee operation in the Kapitza regime [24]. Here we consider the case of $\zeta =20\text{nm}$ (i.e. $\zeta \sim \lambda /30$), but main results remain qualitatively similar for a broad range of modulation periods as long as the condition $\zeta \ll \lambda$ is satisfied. We also fix mean permittivity value ${\epsilon }_{\mathrm{bg}}=0.1$ that leaves modulation depth $p$ as the only control parameter in Eq. (1). It should be stressed that numerical simulations show that increasing ${\epsilon }_{\mathrm{bg}}$ (that can be made comparable with permittivity of some natural materials) requires progressively increasing modulation depths $p$ for the same degree of diffraction suppression for a given light beam. Note that this affords us an additional degree of freedom for the impedance match of the incidence space with that of the Kapitza medium, a condition that has been assumed below. We consider smooth permittivity variations to minimize backward reflections, but the same effect can be observed for step-like $\epsilon (z)$ profiles, as demonstrated in the final part of the paper. Potentially, periodic permittivity landscapes may be fabricated using periodically patterned graphene [27,28], or by stacking metallic and dielectric layers, although in this latter case the scheme will suffer from metallic absorption.

The propagation of the one-dimensional TM-polarized light beams(the only non-vanishing components of the electromagnetic field are $E_x,E_z,H_y$) in the Kapitza medium is governed by the reduced system of the Maxwell’s equations:

We are primarily interested in the impact of the internal beam structure on the rate of its diffraction broadening in the Kapitza medium. We thus use structured beams with the following initial distributions of the magnetic field:

 

Fig. 1 Magnetic field distributions in the input Gaussian (a) multipole beams with $\Omega =0.3\pi /w$ (b) and $\Omega =2.7\pi /w$ (c). The modulus of their spatial spectrums are shown in (d), (e) and (f), respectively. In all cases $w=600\text{nm}$.

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Figure 2 illustrates propagation dynamics of Gaussian beams with widths ranging from $w=40\mathrm{nm}$ (deeply subwavelength regime) to $w=600\mathrm{nm}$ (the borderline regime with $w\approx \lambda$) in the Kapitza medium with a fixed modulation depth $p=1.5$. Notice that, in all cases the propagation distance of $10\mu m$ substantially exceeds the Rayleigh length for the beam of width $w$ defined for the effective permittivity ${\epsilon }_{\mathrm{eff}}={\epsilon }_{\mathrm{bg}}$ and even for ${\epsilon }_{\mathrm{eff}}=\max \epsilon (z)$. The propagation dynamics is drastically different from the case of a uniform medium. For relatively broad (but still nonparaxial) beams rapid longitudinal modulation of the dielectric permittivity leads to nearly complete diffraction cancellation and conservation of the bell-shaped profiles [Figs. 2(e) and 2(f)]. The regime is exceptionally robust in a sense that even moderate fluctuations in the beam shape and modulation depth do not affect notably the width of the output pattern. At the same time, too narrow beams develop two sidelobes gradually expanding in the course of propagation [Figs. 2(a)-2(c)]. Notice, that rapid and deep oscillations of $\epsilon$weakly affect local amplitude of the beam, i.e. it evolves on the scale much larger than modulation period $\zeta$. The angle between two sidelobes increases monotonically with decrease of $w$. Figure 2 shows that the efficiency of diffraction cancellation in the Kapitza medium is determined by the transverse scale of the beam. The impact of $w$ is quantified in Fig. 4(a), where we plot the trapping factor, defined as the ratio $E_{\text{tr}}=U_{out}/U_{\mathrm{in}}$ of the output at $z=3\mu m$ and input $(z=0\mu m)$ powers$U_{\text{in,out}}=(1/2){\displaystyle {\int }_{-w/2}^{w/2}{E}_x{H}_{y}dx}$ concentrated within the transverse window $x\in [-w/2,+w/2]$, as a function of the beam width $w$. This ratio quickly grows with $w$ and approaches unity already for $w\approx 200\mathrm{nm}$.

 

Fig. 2 Evolution dynamics of simple Gaussian beams with width$w=40\mathrm{nm}$ (a),$80\mathrm{nm}$(b),$120\mathrm{nm}$(c),$200\mathrm{nm}$(d),$400\mathrm{nm}$(e), and $600\mathrm{nm}$ (f), in the Kapitza medium with modulation depth $p=1.5$and modulation period$\zeta =30\mathrm{nm}$. Propagation distance is$10\mu m$. Note that in Fig. 2 and all the other following figures for the propagation simulations, the horizontal and vertical windows are not properly scaled in order to reduce the figure size.

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Now we study how efficient the Kapitza medium can arrest the diffraction effects for subwavelength and yet complex structured beams, which is the main goal of the present Letter. The suppression of diffraction for structured dipole and multipole beams upon gradual increase of the permittivity modulation depth $p$in the Kapitza medium, is illustrated in the left and right columns of Fig. 3, respectively. One can see that irrespectively of the structure of the input beam, rapid diffraction at $p=0$ [Figs. 3(a) and 3(f)] is replaced by the formation of two sidelobes at moderate $p$ values [Figs. 3(b) and 3(g)], gradual decrease of the angle between sidelobes, and subsequent transition to nearly diffractionless evolution for large modulation depths [Figs. 3(e) and 3(j)]. For a fixed modulation depth $p$ the beam with more complex internal structure (larger internal frequency $\Omega$) always exhibits faster diffraction broadening. This point can be clearly seen by comparing Figs. 3(e) and 3(i): while a modulation depth $p=1.5$already well eliminates the diffraction effect for dipole beams, a larger depth $p=1.7$ is yet insufficient to counteract diffractions for multipole beams. We further quantify the dependence of the trapping factor $E_{\mathrm{tr}}$on the beam complexity $\Omega$in Fig. 4(b). The monotonic decreasing characteristics of the function $E_{\mathrm{tr}}(\Omega )$ clearly reveals that the efficiency of the Kapitza medium in its diffraction cancellations continuously drops if the beam complexity increases. Still, diffractionless propagation can be achieved for arbitrarily complex input patterns, provided that permittivity modulation depth is large enough [Fig. 4(c)]. Saturation of $E_{\mathrm{tr}}$ at the $E_{\mathrm{tr}}\approx 1$ levels, indicating on the diffraction cancellation, occurs for progressively increasing modulation depths when internal modulation frequency $\Omega$ grows.

 

Fig. 3 Evolution dynamics of multipole beams with $w=600\mathrm{nm}$, $\Omega =0.3\pi /w$ (a)-(e) and $\Omega =2.7\pi /w$ (f)-(j) in the Kapitza medium for progressively growing modulation depth of the dielectric permittivity. Panels (a)-(e) correspond to $p=0,0.4,0.6,0.8$and $1.5$, respectively. Panels (f)-(j) correspond to $p=0,0.5,0.8,1.7,$and $3$, respectively. Propagation distance is $10\mu m$.

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Fig. 4 The ratio of the output and input powers concentrated within $x\in [-w/2,+w/2]$ window (a) versus width of Gaussian beam at $p=1.5$, (b) versus $\Omega$ for complex multipole beam at $p=1.5$ (curve 1) and $p=3$ (curve 2), (c) versus permittivity modulation depth for complex beams with $\Omega =0.3\pi /w$ (curve 1) and $\Omega =2.7\pi /w$ (curve 2). (d) The angle of refraction of tilted Gaussian beam with ${\theta }_{\mathrm{in}}={14}^{\circ }$ versus permittivity modulation depth. In panels (b)-(d) the width of the envelope is $w=600\mathrm{nm}$. Horizontal dashed lines in (a)-(c) correspond to $E_{\mathrm{tr}}=1$ level.

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In all results reported above we assumed that Gaussian or structured beams propagate parallel to the $z$-axis of the Kapitza medium. Importantly, we found that in the case of tilted inputs the longitudinal permittivity modulation not only suppresses diffraction, but also allows to control the propagation direction inside the periodic medium. In order to illustrate this we consider refraction of the beam impinging on the vacuum-Kapitza medium interface at fixed angle ${\theta }_{\text{in}}={14}^{\circ }$ with respect to the $z$-axis. In this case the angle of refraction strongly depends on the permittivity modulation depth $p$ [Fig. 4(d)]. The angle of refraction $\theta$initially grows with $p$ and reaches large values close to $\pi /2$. Further increase of $p$ is accompanied by an abrupt inversion of sign of the refraction angle and its subsequent gradual reduction. At large permittivity modulations the diffraction is suppressed and the refracted beam always propagates parallel to the $z$-axis irrespectively of the value of the input angle ${\theta }_{\mathrm{in}}$. Evolution dynamics illustrating various refraction regimes for progressively increasing permittivity modulation depths is shown in Fig. 5. Such giant and controllable beam refraction at the interface of Kapitza medium, far exceeding that available with shallow refractive index modulation [30], may find potential applications for beam steering and routing at the subwavelength scale.

 

Fig. 5 Propagation dynamics of tilted Gaussian beams with $w=600\mathrm{nm}$,${\theta }_{\mathrm{in}}={14}^{\circ }$ in the Kapitza medium with $p=0$ (a),$0.06$ (b), $0.18$ (c), $0.2$(d), $0.3$ (e), and $1$ (f). Propagation distance is $7.5\mu m$.

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Finally, we note that the Kapitza effects reported above also occur in the layerd metallo-dielectric structures, such as the alternative ultrathin layers of silicon and silvers(Fig. 6(a)) . Figure 6 (d) and 6(e) present a comparative propagation simulation for a dipole and multipole beam input in such a layered Kapitza material; it is evident that the trapping capacity of the Kapitza medium decreases with the increase of the beam complexity (Fig. 6(b)), too. In Fig. 6(c, f, g) we also present the results when silver’s losses is taken into account, showing qualitatively similar pictures as in the lossless case. Especialy, wenote that the diffractionless propagation in the Kaptiza media is not hamperd by the medium losses.

 

Fig. 6 (a) A schematic of the Kapitza structure composed of alternative layers of silicon and silver, with their thickness $d_{Si}=17.87\text{ nm}$and $d_{Ag}=10\text{ nm}$. Dilectric permittivity of silicon is ${\epsilon }_{Si}=12.25$, while that of silve is ${\epsilon }_{Ag}=-20$in (b, d, e) and ${\epsilon }_{Ag}=-20+0.19i$in (c, f, j). (b, c) The ratio of the output and input powers. (d, f) $\Omega =0.3\pi /w$, (e, j) $\Omega =8.3\pi /w$. Propagation distance in (d-g) is $8.0\mu m$.

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3. Conclusion

Summarizing, we showed that, while the rapid large-amplitude modulation of the dielectric permittivity of the material may still suppress diffraction broadening for structured subwavelength beams with arbitrarily complex intensity distributions, the efficiency of the diffraction cancellations lowers down upon the increasing complexity of the beam shape. Small-scale permittivity modulations qualitatively affect refraction at the interface of the Kapitza medium, and sufficiently large modulation redirects all the input beams into the direction of the permittivity of modulations. All these findings were demonstrated in the visible light regime.