## Abstract

We show that diffraction-suppressed propagation of light can be achieved in one-dimensional multilayer metal-dielectric structure, leading to high-resolution imaging through metallodielectric nanofilms.

©2006 Optical Society of America

## 1. Introduction

Wave diffraction is one of the fundamental phenomena associated with propagation of a finite-size beam. Diffraction is a geometric effect that spreads the transverse profile of the beam and thus limits spatial resolution. When the size of the object is comparable to the wavelength of light, the diffractive spreading blurs the image of the object after propagation. In 1987, a class of Bessel-beam solutions of the homogeneous Helmholtz equation was found to be “nondiffracting” [1, 2] in the sense that the central spot of the beam stays localized at the expense of side-lobes that spread with propagation. With recent advances in photonic research there is a growing interest in diffraction management using planar waveguide arrays [3] and photonic crystals [4]–[7]. The existence of photonic bands not only controls transmission frequencies, but also deeply affects the spatial dynamics of the beams, leading to nondiffracting beams [5] and superlensing phenomena in high-dimensional photonic crystals [8]–[9]. Subwavelength super-resolution imaging was also demonstrated with a thin silver film [10]–[13] as an alternative experimental proof of the “perfect lens” [14] concept with left-handed materials (LHM). In the above-mentioned studies, the physical mechanism of superlensing is negative refraction and recovery of evanescent components. Matched impedance is required to minimize the interference with reflection waves. In this paper, we show that nondiffracting beams can also exist in properly designed low-dimensional periodic structures, i.e. 1D multilayer alternating metallic and dielectric (MD) materials, leading to high-resolution imaging through the MD medium. In our case, the physical mechanism of high-resolution imaging is rather distinct from that of the LHM imaging. Here, the high-resolution imaging of the small object is the result of diffraction-suppressed propagation, a consequence of photonic engineering in a multilayer metallodielectric medium. Thus, the matched impedance is not required. By managing photonic bands, we will show that a nearly flat diffraction curve can be designed into a properly engineered metallodielectric medium. Hence, the beam field has little diffraction upon propagation within the MD medium, leading to a higher spatial resolution than would be obtained with propagation in free space of the same distance.

## 2. Diffraction curves

The MD composite consists of alternating layers of dielectric and metallic materials of layer thickness *d*
_{1} and *d*
_{2}, respectively [15]. The metallic and dielectric layers are stacked along the z direction. We assume the permeability is constant (*μ*
_{1} = *μ*
_{2} = 1) for both dielectric and metallic materials and the permittivity is constant for the dielectric material (*ϵ*
_{1}). We use the Drude model for metals with the dispersion given by *ϵ*
_{2}(*ω*) = 1 - *ω _{p}*

^{2}/(

*ω*

^{2}+

*iγω*), where

*ω*is the plasma frequency and

_{p}*γ*is the damping factor. To analyze the effect of photonic bands on the wave diffraction, it is sufficient to consider two-dimensional waves

*E*(

*x*,

*z*) propagating along the

*z*direction. Analogous to dispersion, diffraction is a phenomenon where different spatial frequencies

*k*accumulate different phases upon propagation. In free space, the phase is simply given by

_{x}For multilayer periodic MD medium, the propagation modes are Bloch waves. Thus, the phase accumulation upon propagation is given by *K _{B}*(

*k*)

_{x}*z*, where

*K*is the Bloch wave number that is a function of transverse spatial components. The diffraction relation of the MD medium is a function of the wavelength, material parameters, and thickness of the layers, and can be obtained directly from the dispersion relation [15]:

_{B}where *α _{i}*

^{2}=

*k*

_{x}^{2}- (

*ω*/

*c*)

^{2}

*ϵ*, (

_{i}μ_{i}*i*= 1,2). The existence of the Bloch modes requires that |cos(

*K*)| ≤ 1. Within the MD media, the Bloch modes represent propagation waves when

_{B}d*α*

_{i}^{2}< 0 and evanecent waves when

*α*

_{i}^{2}> 0. The diffraction curve,

*K*versus

_{B}*k*, can be derived directly from Eq. (2). The diffraction relation is much more complicated for the MD medium than that for free space Eq. (1). Our studies show that the diffraction in the MD medium can be either faster or slower than that in free space depending on the parameters. One important feature of the MD stack is that in the absence of material losses, evanscent waves can be coupled and transmitted from the top to the bottom layers regardless of the total thickness of the stack as long as the thickness of each layer is thin enough [15, 19]. The plasma frequency of metals can be as low as 3.5 fs

_{x}^{-1}[16] and as high as 13 fs

^{-1}[17]. Typically, the damping factor is much smaller than the plasma frequency,

*γ*~ 0.01

*ω*[16, 18]. By properly choosing parameters with realistic values, a nearly flat diffraction curve in the MD medium can be achieved. Figure 1 compares the diffraction curve of the MD medium with that of free space. The flat diffraction curve in the MD composite means that the phase accumulation upon propagation has the same rate for different spatial frequencies. Hence, the light propagation in the MD medium experiences much less diffraction than propagation in free space.

_{p}## 3. Diffraction-supressed imaging

Diffraction-suppressed propagation leads to high-resolution imagery. In a linear space-invariant optical system, the wave propagation acts as a spatial filter and is characterized by the optical transfer function (OTF). Let a two-dimensional object be placed at the plane *z* = 0 where the size of the object is comparable to the wavelength of light. Consider the propagation of light from the plane *z* = 0 to a parallel plane at a distance *Z*, the field distribution *U*(*x*,*y*, *Z*) is related to the field at the first plane *U*(*x*,*y*,0) through

where the ℱ represents the Fourier transform, and the *H*(*f _{x}*,

*f*) is the OTF of the propagation from the plane

_{y}*z*= 0 to the plane

*z*=

*Z*. The

*f*and

_{x}*f*are spatial frequencies in the

_{y}*x*and

*y*directions, respectively. In near-field imaging, both propagating and evanescent components must be considered. For free space propagation, the

*H*(

*f*,

_{x}*f*) is simply given by the propagation of the angular spectrum of plane waves including evanescent components. Shown in Fig. 2 is the propagation through the MD medium which has thickness

_{y}*L*. The MD stack is placed at distance

*L*

_{1}from the object plane (

*z*= 0) on the left and distance

*L*

_{2}from the image plane (

*z*=

*Z*) on the right. The total propagation distance

*Z*=

*L*

_{1}+

*L*+

*L*

_{2}comprises two free-space regions and the MD medium. The OTF is given by cascading two free-space propagations and the propagation through the MD stack. The end surfaces of the MD medium are truncated such that the MD stack possesses mirror symmetry [19]. We consider TM polarization since only TM polarization can excite surface plasmons that assist the transmission of high spatial components through the MD stack. For our geometry, there is no coupling between TM and TE polarizations, and the TE polarized fields do not play a role in the imaging process as discussed in this work. The OTF of the MD stack for the TM polarization is computed by using transfer-matrix method (TMM) [15, 19] that includes both propagating and evanescent components. Note that the TMM has built-in vector nature of the electromagnetic fields and the method inherently takes into account multiple reflections at all boundary interfaces. The boundary conditions at all interfaces between metal and dielectric layers are automatically satisfied in the TMM.

To show how the diffraction affects image quality, assume the object comprises two slits of width 1*μ*m and separated by 3*μ*m. Figure 3 compares the intensity distribution at the plane *z* = *Z* with and without the MD stack. In the absence of the MD stack, the image is blurred by the central intensity peak coming from diffraction-induced constructive interference, since the width of the slits broadens due to the diffraction. In the presence of the MD stack, the diffraction is suppressed and the two slits can be clearly resolved. To demonstrate diffraction-suppressed high-resolution imaging, we use a two-dimensional object of letter ‘H’ as shown in Fig. 4. The image of letter ‘H’ after propagating a distance *Z* is given by Fig. 5 with and without the MD stack. In the presence of the MD stack, a clear image is obtained as the result of nondiffracting propagation inside the MD medium. In contrast, the image is blurred after propagating through free space of the same distance.

## 4. Summary

We have shown that diffraction-suppressed high-resolution image can be achieved through the multilayer metallodielectric medium. Nearly diffraction-free beams can exist in properly designed low-dimensional metallodielectric photonic materials. This work suggests that it is possible to provide another degree of freedom for designing high-resolution optical systems to improve diffraction-induced image degradation.

## Acknowledgments

This work is supported by Office of Naval Research (ONR) Independent Laboratory In-house Research and Independent Applied Research funds.

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