Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers
aa r X i v : . [ phy s i c s . op ti c s ] F e b Sub-wavelength diffraction-free imaging with low-lossmetal-dielectric multilayers
R. Kotynski , T. Stefaniuk , A. Pastuszczak
University of Warsaw, Faculty of Physics, [email protected], [email protected], [email protected]
Abstract — We demonstrate numerically the diffraction-free propagation of sub-wavelengthsized optical beams through simple elements built of metal-dielectric multilayers. The proposedmetamaterial consists of silver and a high refractive index dielectric, and is designed using theeffective medium theory as strongly anisotropic and impedance matched to air. Further it ischaracterised with the transfer matrix method, and investigated with FDTD. The diffraction-free behaviour is verified by the analysis of FWHM of PSF in the function of the number ofperiods. Small reflections, small attenuation, and reduced Fabry Perot resonances make it aflexible diffraction-free material for arbitrarily shaped optical planar elements with sizes of theorder of one wavelength.
1. INTRODUCTION
Since the seminal paper by Pendry [1], subwavelength imaging at visible wavelengths has beendemonstrated in much thicker low-loss layered silver-dielectric periodic structures [2, 3, 4, 5, 6, 7,8, 9]. In fact, the effective medium theory (EMT) provides sufficient means to explain enhancedtransmission through the metal-dielectric stacks. Subwavelength resolution results from the extremeanisotropy of the effective permittivity tensor [2, 6, 7].Let us continue the introduction by linking the metal-dielectric multilayers for sub-wavelengthimaging with the concepts taken from Fourier Optics. We refer to the model of a linear shiftinvariant scalar system (LSI) [10, 11] for the description of optical multilayers in a situation whenthey act as imaging nano-elements for coherent monochromatic light. In-plane imaging through alayered structure consisting of uniform and isotropic materials represents a LSI, for either TE orTM polarisations. Linearity of the system is the consequence of linearity of materials and validityof the superposition principle for optical fields. Shift invariance results from the assumed infiniteperpendicular size of the multilayer and the freedom in the choice of an optical axis.A scalar description is valid for the TE and TM polarisations in 2D since all other field compo-nents may be derived from E y or H y , respectively, where the co-ordinate system is oriented as infig. 1.For the TM polarization, the magnetic field H y ( x, z ) may be represented with its spatial spec-trum ˆ H y ( k x , z ) H y ( x, z ) = Z + ∞−∞ ˆ H y ( k x , z ) exp ( ık x x ) dk x , (1)where, at least for lossless materials, the spatial spectrum is clearly separated into the propagatingpart k x < k ǫ and evanescent part k x > k ǫ . The transfer function t ( k x ) (TF) is the ratio of theoutput to incident fields spatial spectra and corresponds to the amplitude transmission coefficientof the multilayer ˆ H y ( k x , z = L ) = t ( k x ) · ˆ H Incy ( k x , z = 0) . (2)Due to reflections, the incident field ˆ H Incy ( k x , z = 0) differs from the total field ˆ H y ( k x , z = 0).The point spread function (PSF) is the inverse Fourier transform of the TF and has the inter-pretation of the response of the system to a point signal δ ( x ). The response to an arbitrary input H Incy ( x, z = 0) can be further expressed as its convolution with the PSF H y ( x, z = L ) = H Incy ( x, z = 0) ∗ P SF ( x ) (3) a. x Ag x Ag x Ag x I N C I D E N C E P L A N E O U T P U T P L A N E L =N · (cid:0)(cid:1) d x /2 d d zx b. Ag x Ag x Ag x I N C I D E N C E P L A N E O U T P U T P L A N E L =N · (cid:2)(cid:3) d d zx c. Effective medium I N C I D E N C E P L A N E O U T P U T P L A N E Lzx
Figure 1: Schematic of a periodic silver-dielectric multilayer with symmetric ( a ) or non-symmetric ( b )composition, and the equivalent effective medium slab ( c ).a. b.Figure 2: a Intensity transmission and reflection coefficients (
T, R ) of the symmetric multilayer as a functionof the filling factor, plotted over the transfer function (in vertical cross-sections of the color map). Phaseisolines are separated by π/
4. The pitch Λ = d + d is fixed at 22 nm , and the total thickness is L = 30Λ. b. Characteristics of the symmetric and non-symmetric multilayers, and of the equivalent slab made of aneffective medium, in the function of the number of periods. Intensity transmission and reflection coefficientscalculated at normal incidence (top); FWHM of
P SF and | P SF | (bottom). PSF of an imaging system usually provides clear information about the resolution, loss or en-hancement of contrast, as well as the characteristics of image distortions. However, for subwave-length imaging, the PSF is not a straightforward measure of resolution and even imaging of objectssmaller than the FWHM of PSF is possible [12].In this paper we demonstrate a diffraction-free material for subwavelength sized optical beams.We combine the following properties of the multilayer: PSF with sub-wavelength size and little de-pendence on the thickness of the structure, high transmission, low losses, and a limited dependenceof the imaging properties on the size of external layers. Together, these properties allow to use amultilayer as a flexible construction material for various optical imaging nano-devices.
2. IMAGING WITH SUB-WAVELENGTH RESOLUTION IN METAL-DIELECTRICMULTILAYERS
The dispersion relation of a two-component infinite stack for the TM polarisation has the form [13] cos( K z Λ) = cos( k z f Λ) cos( k z (1 − f )Λ) − sin( k z f Λ) sin( k z (1 − f )Λ)2 (cid:18) k z ǫ ǫ k z + k z ǫ ǫ k z (cid:19) , (4)where K z is the Bloch wavenumber, Λ = d + d is the period of the stack, d i and ǫ i are the layerthickness and permittivity of material i = 1 ,
2, the filing fraction of material 1 is f = d / Λ and thelocal dispersion relations are k iz + k x = k ǫ i . Wavevector k x is conserved at the layer boundariesand depends on the incidence conditions. The Bloch wavenumber K z is real for a Bloch mode ina lossless stack, however for evanescent waves in a finite stack or for lossy materials K z may becomplex. In the first BZ, the real part of K z satisfies π/ Λ < = re ( K z ) < = π/ Λ. The group velocityas well as the imaginary part responsible for absorption do not depend on the choice of BZ.When the layers are thin K z Λ , k iz Λ <<
1, the second order expansion of (4) over the argumentsof trigonometric functions leads to the dispersion relation for an uniaxially anisotropic effectivemedium K z /ǫ x + k x /ǫ z = k , (5)with ǫ x = f ǫ + (1 − f ) ǫ and ǫ z = 1 / ( f /ǫ + (1 − f ) /ǫ ). This is the basis of the effective mediumapproximation thoroughly discussed by Wood et al. [2], also as the basis for applying the near-field approximation. The transmission coefficient of the Fabry-Perot (FP) slab consisting of thehomogenized effective medium is then given as [2], t ( k x ) = (cos( k z L ) − . ı ( K z ǫ x /k z ǫ + k z ǫ/K z ǫ x ) sin( k z L ))) − , (6)where k z and ǫ refer to the external medium, and L is the total thickness of the slab. At thesame time t ( k x ) is the already mentioned coherent amplitude transfer function of the imagingsystem [11]. When | ǫ x /ǫ z | << ǫ /ǫ = − d /d [7]), forcertain slab thickness the resonant FP condition becomes independent on the angle of incidence.This happens when L = λm/ √ ǫ x , m ∈ N . Then t ( k x ) ≡ exp ( − ıK z L ) and the FP slab introducesthe same phase shift for all harmonics of the spatial spectrum. Belov and Hao [7] proposed tocombine this condition with impedance matching between the external medium and the effectiveFP slab ǫ = ǫ x = f ǫ + (1 − f ) ǫ and referred to that regime as canalization. However, Li et al. [8]questioned the importance of impedance matching in favour of the FP resonance condition. In fact,for a lossless metal and dielectric, the FP resonance is sufficient to entirely eliminate reflectionsresulting in perfect imaging t ( k x ) ≡ ǫ = ǫ x only approximate and, at the same time, the finite value of Λ limits the validityof homogenisation. Moreover, transmission through a finite slab strongly depends on the materialand thickness of the external layers and appears to be the largest for a symmetrically designedslab (Fig. 1a) with half-width dielectric layers located at the boundaries [3].
3. NUMERICAL DEMONSTRATION OF DIFFRACTION-FREE PROPAGATION OFSUB-WAVELENGTH SIZED OPTICAL BEAMS
After recalling the theory of sub-wavelength imaging in metal-dielectric multilayers, we now focuson an imaging regime which may be called diffraction-free. We note, that the FP resonances areaccompanied with a field pattern inside the slab similar to a standing wave [2, 9]. Therefore lookfor structures for which t ( k x ) ≈ const nonetheless FP resonances are weak.Let us now focus on an example of a metal-dielectric periodic multilayer consisting of silver andhigh refractive index dielectric, which enables us to demonstrate and explain the diffraction-freepropagation of sub-wavelength sized optical beams. The structure operates at the wavelength of λ = 422 nm , when the permittivity of silver is equal to ǫ = − . . ı [14], and the permittivityof the dielectric such as T iO or SrT iO [10, 15] is ǫ = (2 . . Layer thicknesses are assumedto be d = 10 nm , d = 12 nm with periodic symmetric or non-symmetric composition shown infig. 1a,b. The fraction d /d may be justified with the use of EMT. The corresponding permittivityof the effective medium gives ǫ x ≈ .
02 + 0 . ı and ǫ z ≈ −
158 + 191 ı , which assure impedancematching with air ǫ = 1 ≈ ǫ x together with the condition for the extreme anisotropy | ǫ x /ǫ z | << n x = √ ǫ x is reduced by the factor of 50 compared tobulk silver, resulting in low-loss transmission. In fig.2a we show the transfer function calculatedrigorously with TMM for a range of filling factors 0 < d / Λ < .
6, when the total thicknessof the structure is fixed at L = 660 nm . Indeed, the impedance matching which occures when a. z / λ x / λ Poynting vector S z [a.u.] b. z / λ x / λ Instantanous distribution of Hy
Hy[a.u.] −0.500.5 c. z / λ x / λ Instantanous distribution of Hy
Hy[a.u.] −0.6−0.4−0.200.20.40.6
Figure 3: Diffraction-free transmission of the subwavelength sized beam through the multilayer (FDTDsimulations, with the incident CW beam limited by a subwavelength aperture in a perfect conductor): a . time-averaged Poynting vector S z in a rectangular slab; b,c . instantaneous magnetic field H y in a slabwith layer boundaries oriented at 30 ◦ towards the external boundaries for various aperture sizes.a. z / λ x / λ Instantanous distribution of Hy
Hy[a.u.] −0.6−0.4−0.200.20.40.6 b. z / λ x / λ Instantanous distribution of Hy
Hy[a.u.] −0.500.5
Figure 4: Simple optical nano-elements internally made of the diffraction-free material a . double slab; b .prism;. d = 10 nm, d = 12 nm results in reduced reflections, high transmission of both propagating andevanescent spatial harmonics, and a flat phase of the transfer function for a broad range of k x /k .The size of the corresponding PSF is of the order of λ/ ≈ nm assuring the super-resolvingproperties of the device. Fig. 2b shows the FWHM of PSF alongside with the reflection andtransmission coefficient in the function of number of periods, for the symmetric, non-symmetric andeffective-index composition of the structure. The major properties of the structure are the following:the size of PSF varies slowly with the size of the structure and is almost the same for the symmetricand non-symmetric composition, the FP resonances observed in transmission are weak (as opposedto those observed in reflections), the attenuation is uniform as the number of layers is increased.These properties assure an almost diffraction-free propagation through the structure, with a similarattenuation and the size of PSF for symmetric and non-symmetric composition and for a broadrange of structure sizes L . This behaviour is illustrated in fig. 3a with an FDTD [16] simulationshowing the uniform non-diverging distribution of the Poynting vector inside the structure for abeam size of the order of λ/
10. Transmission takes place along the direction normal to the layerboundaries with negligible divergence. Furthermore, our material may be used to fabricate slabs cutat arbitrary angle to layer surfaces. As is shown in fig. 3b,c we continue to observe the diffraction-free propagation for inclined layers, whether the aperture covers only a single or multiple periodsof the grating. This may be heuristically explained with the similar PSF for a symmetric and non-symmetric composition of multilayers. Both of them are encountered at the cross sections drawnalong the propagating beam, starting from various points inside the aperture and normal to thelayer boundaries. Furthermore, it is possible to construct other simple optical nano-elements. Infig. 4 we demonstrate the operation of a double multilayer, which may be part of a cloaking deviceor an optical interconnect, as well as prism for imaging subwavelength beams.
4. CONCLUSIONS
We have demonstrated a diffraction-free, low-loss material, which is impedance matched to air.The diffraction-free propagation is verified by the analysis of FWHM of PSF in the function of thenumber of periods. The material consists of silver and a dielectric with refractive index of n = 2 . λ = 422 nm . It has the effective imaginary part of refractive index smaller thanthat of silver by a factor of 50. The reflections are in between − dB and − dB , and the FWHMof PSF is at the order of 40 nm . Small reflections, small attenuation, and reduced Fabry Perotresonances make it a flexible metamaterial for arbitrarily shaped optical planar devices with sizesof the order of one wavelength, such as the elements of optical interconnects or cloaks. ACKNOWLEDGMENT
We acknowledge support from the Polish MNiI research projects
N N202 033237 and
N R15 001806 as well as the framework of COST actions
MP0702 and
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