## Abstract

The Talbot effect (or the self-imaging effect) can be observed for a periodic object with a pitch larger than the diffraction limit of an imaging system, where the paraxial approximation is applied. In this paper, we show that the super Talbot effect can be achieved in an indefinite metamaterial even when the period is much smaller than the diffraction limit in both two-dimensional and three-dimensional numerical simulations, where the paraxial approximation is *not* applied. This is attributed to the evanescent waves, which carry the information about subwavelength features of the object, can be converted into propagating waves and then conveyed to far field by the metamaterial, where the permittivity in the propagation direction is negative while the transverse ones are positive. The indefinite metamaterial can be approximated by a system of thin, alternating multilayer metal and insulator (MMI) stack. As long as the loss of the metamaterial is small enough, deep subwavelength image size can be obtained in the super Talbot effect.

© 2011 OSA

## 1. Introduction

Advanced developments of metamaterial provide the opportunities to independently control the permittivity or permeability of material as well as to engineer either of the two macroscopic parameters to be almost any arbitrary value (positive, zero or negative) [1–4], which bestow electromagnetic properties unattainable in the natural-occurring material. Recently, some research shows that an anisotropic metamaterial, in which one of the components of the permittivity tensor has a different sign to the others, exhibits significant advantages over conventional material. This type of anisotropic material is called as indefinite metamaterial since the permittivity components have different signs. In a two-dimensional (2D) anisotropic metamaterial (assuming the material is nonmagnetic), the dispersion relation of light can be expressed as

where*k*,

_{x}*k*, and

_{z}*k*

_{0}are the wave vectors in transverse direction

*x*, propagation direction

*z*and in free space, respectively. From Eq. (1) we have

When *ε _{z}* < 0 and

*ε*> 0, there is no cutoff for any frequency

_{x}*k*in the metamaterial. In other words, the indefinite metamaterial can convert the evanescent waves which would normally decay in conventional material into propagating waves. Compared with the circular dispersion relation in natural material (for example, air), the dispersion curve for an indefinite metamaterial is in hyperbolic form, where

_{x}*k*does not have an upper limit and can vary in (-∞, + ∞). In particular, when$\left|{\epsilon}_{z}\right|\gg 1$, self-collimation of waves can be achieved [5].

_{x}In this paper, we re-visit one classical optical phenomenon, named Talbot effect [6], and investigate this effect in an indefinite metamaterial using two-dimensional (2D) and three-dimensional (3D) finite-difference time-domain (FDTD) methods [7,8]. The self-imaging phenomenon or the Talbot effect is a direct result of Fresnel diffraction [9], which can be observed for a periodic object when illuminated by a monochromatic light. In the transversal direction, the field amplitude is periodic and it is also periodic in the light propagation direction behind the object. Because of its simple arrangement, the Talbot effect attracts researchers’ interests and the self-imaging phenomenon has a variety of applications in the optical dispersive fiber system [10,11], optical computing [12], phase locking of laser arrays [13], and in electron optics and microscopy [14]. The Talbot effect has also been demonstrated with atomic waves [15], and waveguide arrays [16,17]. More recently, the nonlinear Talbot effect was experimentally observed in nonlinear photonic crystals [18]. However, most of the works above are based on the paraxial approximation of the classical Talbot effect. In this work, we will show that super Talbot effect can be observed *without using the paraxial approximation*.

## 2. Super Talbot effect demonstrated in 2D simulations

#### 2.1 Super Talbot effect in an indefinite metamaterial

Figure 1(a)
schematically illustrates the structure we investigated in the numerical simulations. A TM-polarized plane wave (magnetic field is in the *y*-direction) with a wavelength of *λ*
_{0} propagates along *z*-axis and impinges a one-dimensional grating from the left side. The period of the grating is *D*, and the duty cycle is expressed as *d/D*. In the simulation, the period of the grating *D* = 100nm, *d* = 50nm and the incident wavelength *λ*
_{0} = 630nm. In a conventional material, the evanescent waves which carry the information about the subwavelength features of the object will decay exponentially. Hence, if the period of the input object is much smaller than the incident wavelength, no Talbot effect can be seen in the regular material (as shown in Fig. 1(b), in the air). In contrast to the case in the air, self-imaging effect can be achieved in an indefinite metamaterial, which is due to the evanescent waves being converted into propagating waves and conveyed far away. This self-imaging effect observed in the indefinite metamaterial when the period of the object much smaller than the incident wavelength is defined as “super Talbot effect” in this paper. Figure 1(c) clearly shows the strong, periodic magnetic field distribution pattern in the indefinite metamaterial (with *ε _{z}* = −1 and

*ε*= 1) behind the input grating. This effect is related to but also distinct from the plasmon Talbot effect reported in recent work [19,20], where

_{x}*surface*plasmon Talbot effect is observed on the

*surface*of metal and its resolution is still limited by the surface plasmon diffraction limit. In contrast, our super Talbot effect is based on

*bulk*plasmons without diffraction limit.

Since the wavelength *λ*
_{0} is much larger than the period *D*, the paraxial approximation cannot be applied any more. In our case, light propagates in the indefinite metamaterial behind the input grating structure can be expressed in Fourier series form as

*k*is expressed by Eq. (2). Replacing

_{z}*k*by

_{x}*mq*in Eq. (2) and then plugging Eq. (2) into Eq. (3), we get

_{x}Under long wavelength approximation, ${\lambda}_{0}\gg D$and hence${q}_{x}/{k}_{0}\gg 1$, so Eq. (4) can be further simplified as

From Eq. (5) we can get the Talbot distance in an indefinite metamaterial can be expressed as

which only depends on the permittivity components of the metamaterial and the input object period. Note the permittivity of a metamaterial is highly frequency-dependent. Using Eq. (6), the Talbot distance in the indefinite metamaterial (with*ε*= −1 and

_{z}*ε*= 1) can be calculated approximately as 100nm, while the Talbot distance measured in the simulation is around 117 nm. The difference may come from that the ratio of the wavelength

_{x}*λ*

_{0}and the period

*D*is only around 6.3 while Eq. (6) is based on the condition that${\lambda}_{0}\gg D$.

To better understand the super Talbot effect, we also modeled the self-imaging of a large-pitch mask (*D* = 2*µ*m and *d* = 100nm) in a conventional medium and in an indefinite metamaterial, respectively. As shown in Fig. 2(a)
, the Talbot effect can be observed in a conventional material (for example in the air) under paraxial approximation, where the incident wavelength is still 630nm. If the medium behind the grating is replaced by an indefinite metamaterial with *ε _{z}* = −1 and

*ε*= 1, the Talbot carpet pattern still can be clearly observed (shown in Fig. 2(b)). The feature size of the squared hot spot in Fig. 2(b) along the transverse direction measured is approximately as 160nm, which is much smaller than that in the air, 240nm. The result is within our expectation since the high spatial-frequency waves can be conveyed to far field in the indefinite metamaterial, as mentioned previously.

_{x}#### 2.2 Super Talbot effect in a multilayer metal-insulator (MMI) stack

The anisotropic metamaterial can be approximated by a system of thin, alternating multilayer metal-insulator (MMI) stack. As one type of simple metamaterial, the MMI stack has been used for superlens [21–23], optical lithography [24], and subwavelength sensing/detecting [25]. As long as the thickness of each layer is sufficiently thin, the effective medium theory (EMT) [26] can be applied to describe the MMI stack by a macroscopic parameter, the permittivity tensor [26,27]. The effective permittivity tensor of the MMI stack can be determined by:

*η*is the thickness ratio of the two layers and defined by $\eta ={d}_{m}/{d}_{i}$,

*ε*and

_{m}*ε*are the permittivity of the metal and the insulator, respectively. By carefully choosing two suitable materials and a thickness ratio

_{i}*η*,

*ε*

_{Π}and

*ε*

_{⊥}could have opposite signs.

In this work, Ag and SiO_{2} were selected as the metal and insulator, respectively. The structure we investigated is similar as shown in Fig. 1(a), where the material behind the grating was replaced by the MMI stack composed of Ag and SiO_{2} thin layers and they were assumed to be infinite in *y*-axis. The alternating layers were stacked in the *x* direction and each layer has a thickness of 5nm. In the simulations, periodic boundary conditions were implemented for the boundaries parallel to the *z*-axis. The wavelength of the incident TM-polarized plane wave *λ*
_{0} = 630nm, the period of the grating *D* = 94nm and the duty cycle = 50%. At the wavelength of interest, the permittivity of Ag is *ε*
_{Ag} = −15.69 + *i*1.06 and that of SiO_{2} is *ε*
_{SiO2} = 2.12. As *η* = 1, from Eq. (7) the effective permittivity of the MMI stack is calculated to be *ε _{z}* = −6.79 +

*i*0.53 and

*ε*= 4.9 +

_{x}*i*0.05. Even the material loss is considered, the super Talbot effect is still obvious as shown in Fig. 3(a) . Figure 3(b) plots the cross-sectional profile of the power in

*z*-axis (where

*x*= 52nm, the horizontal white solid line in the figure). Each of the peaks numbered from 1 to 7 of the curve represents the location of one Talbot imaging plane where the self-imaging phenomenon occurs. The intensity gradually decays due to the attenuation of the MMI stack. The distance between the two adjacent peaks is the Talbot distance

*Z*

_{T}, which is approximately measured as 152nm. The Talbot distance calculated by Eq. (6) is around 110nm. Both the effective medium theory and the long wavelength approximation may contribute to the difference. If the attenuation of the MMI stack is negligible, deep subwavelength image size can be achieved in the super Talbot effect. At point 1, where

*z*= 40nm (along the first dashed vertical line as shown in Fig. 3(a)), the full-width half-maximum (FWHM) of one image hot spot is measured as 35nm (or 0.056

*λ*

_{0}). At another Talbot imaging plane, where

*z*= 642nm (point 5 in Fig. 3(b), along the second dashed vertical line in Fig. 3(a)), a subwavelength image size of 55nm (or 0.087

*λ*

_{0}) still can be achieved. Comparing the cross-sectional power profiles of point 1 and point 5 (which are not shown in the paper), the contrast of the power at point 5 is not as sharp as that of point 1 and the quality of the image is worse, which are attributed to the loss of high-order diffraction light in the MMI stack.

## 3. Super Talbot effect demonstrated in 3D simulations

When considering the realistic applications, for example in nanolithography, where usually the pattern on the mask in the *y*-axis is also periodic, the super Talbot effect in a 3D configuration needs to be explored. Here we mimic a simple scenario in nanolithography, where light is incident from a chrome (Cr) mask with periodic subwavelength holes, as shown in Fig. 4(a)
. The incident plane wave is polarized in *x*-axis and has a wavelength of 630nm. The parameters of the hole-array are as below: hole diameter 2*r* = 80nm and the periods along the *x* and *y* axes are identical as *D* = 150nm. For simplicity, behind the Cr (*ε*
_{Cr} = −6.3 + *i*31.2) mask is an indefinite metamaterial which is assumed to be lossless and the permittivity components are *ε _{z}* = −4, and

*ε*=

_{x}*ε*= 1. The origin of the whole system is defined at the center of the hole (marked with the red solid circle) with

_{y}*z*= 0 at the interface between the Cr mask and the indefinite metamaterial.

In Fig. 4(b), the |E|^{2} distribution of the vertical *z*-*y* plane at *x* = 0 is depicted, and similarly, Fig. 4(c) for the horizontal *z*-*x* plane at *y* = 0. Clearly, the super Talbot effect can be obtained in both the vertical and horizontal planes. The Talbot distance *Z*
_{T} (as shown in Fig. 4(c)) measured is about (300 ± 15) nm, which agrees with the result from Eq. (6), calculated as 300nm. The asymmetric pattern distributions as shown in Fig. 4(b) and 4(c) may be attributed to the polarization dependent effect. Figure 4(d) shows the image in one of the Talbot imaging planes (*x*-*y* plane, *z* = 365nm). The positions of the hot spots coincide with those of the periodic holes on the Cr mask. The size of each hot spot (FWHM) is approximately 40nm-by-40nm (or 0.0635*λ*
_{0}-by-0.0635*λ*
_{0}). Besides the integer self-imaging planes, we also observed some fractional Talbot imaging planes which locate between two adjacent integer Talbot planes [28], with *z* = *p*/*qZ*
_{T} (*p* and *q* are prime integers). One representative fractional Talbot imaging plane with *z* ≈2/3*Z*
_{T} was shown in Fig. 4(e). Comparing with the integer Talbot imaging plane [Fig. 4(d)], the image on the fractional Talbot imaging plane as shown in Fig. 4(e) has the same spatial frequency but is shifted about 0.5 periods (75nm) along *x*-axis. The image size (FWHM) of those hot spots is measured as 30nm-by-40nm (or 0.0476*λ*
_{0}-by-0.0635*λ*
_{0}). Low loss 3D indefinite metamaterials may be constructed by an array of aligned metallic nanowires (*ε _{m}* < 0) embedded in a dielectric host (

*ε*> 0). The fabrication of the nanowire-based metamaterial is based on a well-developed technique named “template synthetic method” [29–32] and with the pores filled by metals.

_{d}## 4. Conclusion

In conclusion, we have shown that super Talbot effect can be observed in an indefinite metamaterial without the paraxial approximation. A 2D indefinite metamaterial can be approximated by a multilayer metal-insulator stack for super Talbot applications. As long as the loss of the metamaterial is small enough, a deep subwavelength resolution (~0.087*λ*
_{0}) can be achieved. The super Talbot effect may find a variety of applications in the fields as nanolithography and optical storage. In particular, 3D photonic crystals may be fabricated based the super 3D self-imaging phenomenon [33].

## Acknowledgments

This material is based upon work supported in part by the U.S. Army under Award No. W911NF-10-1-0153 and the National Science Foundation under Award No. ECCS-1057381. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research.

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