Unidirectional SPP excitation at asymmetrical two-layered metal gratings

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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 (8pp) doi:10.1088/0953-4075/43/18/185401

Unidirectional SPP excitation atasymmetrical two-layered metal gratingsA Roszkiewicz and W Nasalski

Institute of Fundamental Technological Research, Polish Academy of Science, Adolfa Pawinskiego 5b,02-106 Warsaw, Poland

E-mail: arosz@ippt.gov.pl

Received 22 March 2010, in final form 19 July 2010Published 6 September 2010Online at stacks.iop.org/JPhysB/43/185401

AbstractA new configuration of a reflective one-dimensional (1D) surface-relief metallic-gratingstructure for unidirectional excitation of surface plasmon polaritons (SPPs) is proposed andnumerically explored for plane wave normal incidence. The structure is embedded betweentwo different dielectric media and composed of two layers, each of them consisting ofperiodically placed rectangular metal stripes. It is shown that even a small horizontal shiftbetween these two layers, or a change in dielectric contrast of the grating fillings, may redirectenergy flow propagation in the vicinity of the grating structure. From these two cases ofcounter propagating energy flows, only one shows the field behaviour typical of SPPexcitation. This evident directivity phenomenon exists together with a high concentration ofthe electromagnetic field at the periodic structure. The configurations analysed may be usefulin designing optical devices like photodetectors or, in general, in any case where efficientcontrol of energy propagation directivity is of primary importance.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Plasmonics is one of the most rapidly developing branches ofmodern science, with tremendous progress in all its theoreticaland experimental aspects [1]. Great effort has recently beenmade in developing efficient methods for controlling surfaceplasmon polaritons (SPPs) [2] and designing new plasmonicdevices, with a growing number of potential applications inseveral aspects of science and technology [3]. These include,among others, spectroscopic chemical and biological sensorsused in industrial process control, monitoring environmentalpollution and planet surfaces, remote sensing with opticalfibre sensors, and production of efficient solar cells andlight-emitting diodes with high efficiency. Due to thestrong wavelength discrimination effect, SPPs also allow theproduction of wavelength-tunable filters. The ability of SPPto overcome the Rayleigh criterion together with a greatenhancement of the electromagnetic field in the vicinity ofa metal surface allow the visualization of nano-objects ofdimensions much smaller than an incident wavelength. Inoptical communication and data storage, SPPs are consideredto be a new type of optical information carrier.

For the further research and development of plasmonics,devising photonic elements similar to those used inconventional optics, but working with the help of SPPs,is necessary. For this goal, special photonic elementshave been developed, such as bent waveguides with hightransmission efficiency, microgratings working as Braggwaveguides with orthogonal bends, beam splitters, Braggmirrors, interferometers and SPP semiconductor lasers. In thispaper we present another photonic structure which stronglyinterrelates its geometrical parameters with the directivity ofthe SPP propagation.

2. Grating geometry and numerical modelling

SPPs, being electromagnetic waves coupled to the densitywaves of free electrons in metals, are surface waves thatpropagate along the interface between two media with theirdielectric constants of opposite signs [4]. They can be excitedonly with a TM polarized wave and, as having larger wavevector components in the horizontal direction than those ofincident light, it happens only when this wave vector isincreased in the direction of SPP propagation. Commonly used

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

configurations for this purpose are glass prisms or diffractiongratings. In order to achieve SPP propagation in one desirabledirection, an oblique incidence regime should be applied.However, in several configurations, due to the complexity ofthe optical system, it is necessary to excite SPPs by normallyincident beams. This can also be accomplished by usingdiffraction gratings.

When light of frequency ω impinges on a one-dimensional(1D) diffraction grating at an angle θ i , the horizontal (parallelto the grating vector) components of diffracted orders arekx = ωn sin(θ i)/c ± mK, where n is the refractive index ofthe medium through which the grating is illuminated, m is aninteger denoting the diffraction order and c is the speed oflight in vacuum. K = 2π/� is the reciprocal 1D lattice vector,where � denotes the grating period. However, a significantwaste of energy occurs when a symmetric grating is used dueto equal coupling strengths of SPP propagating in the twoopposite directions. This can be prevented by using a non-symmetric grating [5–7]. The authors of [5] analyzed twoaluminium gratings. One was slanted rectangular grating andthe second had a smooth sinusoidal slanted shape. Althoughboth gratings were shown to be able to efficiently lead SPPsin one direction, they are relatively difficult to manufacture,especially in the visible region, due to the requirement ofmaintaining high accuracy, e.g. in preserving slant angle.

Asymmetry of the SPP coupling is also used in designingdirectional beaming effects [8, 9]. Off-axis directionalbeaming originates from the different coupling efficiencies ofdifferent gratings placed on both sides of a subwavelengthslit in an optically thick metal film. By tuning gratings’parameters, such as a grating period in the case of a full-metallic grating or a refractive index in the case of ametallic grating covered by a dielectric layer, it is possibleto adjust precisely an angle of the beam propagation direction.Similarly, breaking the configuration symmetry was utilizedin designing slanted grating directional couplers [10]. Devicesof such a type couple the light coming from a vertical fibredirectly into a specially designed core and further, without anyadditional optics, into a horizontal waveguide.

We propose a different asymmetrical reflectionconfiguration of the diffraction grating for the unidirectionalexcitation of SPPs. By manipulating with the asymmetryof the structure, it is possible to change the direction of alocal energy flow that is manifested by the sign change ofthe local Poynting vector near the surface. We show that itis possible to achieve distinct and measurable effects resultedfrom the asymmetry of the periodic structure. The structureproposed in this paper could be manufactured with the standardphotolithography or focused ion beam milling processes. Inthis configuration, the precise control of depth variation withinthe grating period is not so crucial as for the slanted gratingsmentioned in [5]. In that case, the evaporation of subsequentlayers must be repeated several times with different masks inorder to approximate a desired smooth shape of the grating.

The problem under consideration is depicted in figure 1.The normally incident plane wave of wavelength λ and TMpolarization impinges on the 1D asymmetrical planar periodicstructure. The structure is composed of two metal subgratings,

Figure 1. Geometry of a two-layered grating diffraction problem.Two 1D surface-relief metallic diffraction gratings with differentdielectric media in the gaps between the metal stripes are stackedand shifted in relation to each other.

one on the top of another and shifted in relation to each other.By dx we denote a shift between the centres of metallic stripesin both gratings. A single grating layer is composed of periodicmetal and dielectric areas. A complex-valued refractive indexof metal is described by the Drude model:

n2(ω) =√

1 − ω2p

ω2 + iγω(1)

with parameters for gold: ωp = 1.38 × 1016 s−1 and γ =1.07 × 1014 s−1. Dielectrics placed between metal stripes inthe upper and lower grating layers have the refractive indices n4

and n5, respectively. The dielectric substrate has the refractiveindex n3. The filling factors (areas occupied by metal in eachlayer) are denoted as f 1 and f 2, respectively, and the period ofthe grating is �. The upper medium is air with n1 = 1, althoughmay be replaced, in general, by any constant value n1, when theratio n1/n3 is preserved. Therefore, this configuration shouldbe considered as the metal grating embedded between twodielectric media. The height of the whole grating structureis h, the heights of individual grating layers are h1 and h2,respectively.

In numerical calculations, the rigorous coupled-waveanalysis [11, 12] (also called the Fourier modal method)with factorization rules described in [13] is applied. Thisfrequency-domain method is widely used in grating diffractionanalysis. It is based on the Fourier expansion of the relativepermittivity function in the grating region. Maxwell equationsin the periodic region are solved for the electric and magneticfields decomposed into Floquet–Bloch modes. Eigenvaluesand eigenmodes associated with grating modes are computedfor each grating layer, invariant in the z-direction.

Those solutions are matched with each other andwith solutions in the superstrate and the substrate byboundary conditions imposed on tangential field components.Eventually, the set of equations is solved for transmitted,reflected and grating mode amplitudes.

The energy flow (denoted by arrows on each fourthsubplot of figures 2–5) is expressed by the time-averagedPoynting vector, 〈S〉 = Re(E × H∗)/2, where E and H are theelectric and magnetic field vectors and ∗ denotes the complexconjugation. The Poynting vector in each figure must beconsidered as a resultant energy flow of the incident waveand reflected wave, where the reflected wave may acquire

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

(a) (b)

(c) (d)

Figure 2. Grating configuration with n4 = 1, n5 = 1.5 and dx = 0.2� and all nonzero components of the electromagnetic field, normalizedto the incident wave, in the vicinity of diffraction grating: (a) Ex , (b) Hy and (c) Ez, and (d) the Poynting vector magnitude |S| with thedirection of the energy flow denoted by arrows.

(a) (b)

(c) (d)

Figure 3. Grating configuration with n4 = 1, n5 = 1.5 and dx = 0.375� and all nonzero components of the electromagnetic field,normalized to the incident wave, in the vicinity of diffraction grating: (a) Ex , (b) Hy and (c) Ez, and (d) the Poynting vector magnitude |S|with the direction of the energy flow denoted by arrows.

one from the two (positive or negative) propagation directionsalong the x-axis. The two waves propagating in the twoopposite directions possess, in general, different complex

amplitudes. A dispersive material (gold in our case) hasthe non-zero frequency-dependent absorption coefficient thatincreases significantly in abnormal dispersion regions. The

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

(a) (b)

(c) (d)

Figure 4. Grating configuration with n4 = 1.5, n5 = 1.5 and dx = 0.4� and all nonzero components of the electromagnetic field, normalizedto the incident wave, in the vicinity of diffraction grating: (a) Ex , (b) Hy and (c) Ez, and (d) the Poynting vector magnitude |S| with thedirection of the energy flow denoted by arrows.

(a) (b)

(c) (d)

Figure 5. Grating configuration with n4 = 1, n5 = 1 and dx = 0.25� and all nonzero components of the electromagnetic field, normalized tothe incident wave, in the vicinity of diffraction grating: (a) Ex , (b) Hy and (c) Ez, and (d) the Poynting vector magnitude |S| with thedirection of the energy flow denoted by arrows.

relation between the real and imaginary parts of susceptibilityand the refractive index and the extinction coefficient are givenby

n − iγ

2k0=

√1 + χ1 + iχ2, (2)

where n and γ are the refractive index and the extinctioncoefficient, respectively, and k0 is a free-space wave-number.Kramers–Kronig relations relate n and γ for real (χ1) andimaginary (χ2) parts of susceptibility for a sufficiently widerange of frequencies.

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

In numerical simulations, we assume λ = 632.8 nm,n1 = 1, n3 = 1.5, � = 600 nm, h1 = h2 = 60 nm andf 1 = f 2. The height of the grating should be preciselytailored in order to obtain the directionality effect, as willbe described below. The values of the refractive index ofdielectric media are not considered as absolute values butrather as contrast values with respect to the upper dielectricmedium. For simplicity, we assume that n1 = 1 in the incidentmedium. Different configurations of the two-layer structure,with different directions of the field energy flow, will bepresented in the next section.

3. Asymmetry of the grating configuration and fielddistribution

The grating period is chosen smaller than the wavelength λ inorder to make all diffraction orders, except the zero-order one,evanescent with almost completely suppressed energy losses.When the reflectivity and transmittivity reach their minimumpossible value, the electromagnetic field obtains its maximumvalue near the surface. In order to achieve the propagation ofSPP in only one direction, amplitudes of specularly reflectedwaves have to be significantly decreased, and the energy flowhas to be transferred into the first diffraction order due tothe coupling of the one from many evanescent waves to SPP.Such a wave field should match the following conditions: theequality of the frequency of the incident wave and SPP, aswell as the equality of the incident wave vector horizontalcomponent and the wave number of the SPP. We first optimizedthe filling factors f 1, f 2 of both layers following the proceduresimilar to that described in [5]. The resultant filling factors aref 1 = f 2 = 0.4. The fourth subplots of figures 2–5 present themain result of numerical simulations, that is, the magnitudeand direction of the Poynting vector. The three remainingsubplots displaying the electromagnetic field components Ex ,Hy and Ez indicate the asymmetry of the field distributions.

3.1. Influence of horizontal shifts in the grating geometry

We first considered two structures characterized by therefractive indices n4 = 1 and n5 = 1.5, with the horizontalshift change between the upper and lower layers within therange between 0 and 0.5�. We plotted two configurationswith different shifts: dx = 0.2� (figure 2) and dx = 0.375�

(figure 3). In both figures the plane boundary between twodielectric fillings is placed in between two grating sublayers.

In both cases the asymmetrical distribution of the Poyntingvector magnitude in the vicinity of the structure is clearly seen.There is a small leakage of radiation into the substrate anda distinct directionality of energy flow near the surface. Theshift difference between both configurations is relatively smallbut, nonetheless, it completely changes the direction of energypropagation. Such a small shift difference causes a measurablechange in the propagation conditions and therefore in thedirection of energy propagation in the vicinity of the surface.It can easily be seen that the component of the electromagneticfield that changes its sign at a certain value of dx , reversingtherefore the sign of the Poynting vector, is Hy . Resonance

conditions at the upper and lower boundary are not the samedue to the presence of media with different refractive indicesand, therefore, the SPPs are excited only at one boundary.

The grating period has to be properly chosen with respectto the wavelength. Although, for brevity, it is not shown,we verified that a too short or a too large grating periodwith respect to the wavelength does not allow guiding theelectromagnetic field along the structure in the form of SPP,since the coupling conditions between the incident evanescentorder and the SPP are not fulfilled. This means that for a givenstructure, even asymmetric, the choice of the wavelength rangeis limited by the structure ability to change the direction of SPPpropagation. Additionally, if the grating is too thin or too thick,the directivity effect is also suppressed. This is the reason forchoosing the height of the whole grating equal to 120 nm. Inparticular, the values of half or twice the chosen height (60 nmor 240 nm) are too small or too large to enable a relevantchange in the direction of energy propagation in the describedconfiguration. Only the narrow range of grating heights allowsone to achieve the opposite energy flow directions for the shiftsin the range 0 � dx � 0.5�.

There are also some general restrictions regarding thefilling factor. It cannot be too small in order to preventdirect transmission of light between the metal stripes into thesubstrate. On the other hand, we have also numerically verifiedthat too wide metal stripes cause the creation of additional curlsabove the grating due to the reflection of significant amountof electromagnetic field from metal areas. The reflected fieldinterferes with the incident light and forms an interferencepattern. This phenomenon smears out the directionality andsmoothness of the energy flow.

3.2. Influence of the dielectric medium inside the gratingregions

Subsequently, we analyze the field behaviour above thestructure specified by a vertical shift of the dielectric boundaryinside the grating region. Two cases are presented. In the firstcase, the grating filled with the dielectric of the refractiveindex higher than that of the incident medium n4 = n5 > n1 isconsidered (figure 4). In the second case, the grating with thedielectric medium of the refractive index equal to the indexof the incident medium n4 = n5 = n1 is presented (figure 5).In both cases it is possible to find a particular value of thehorizontal shift (dx) at which the strong directional effect isobtained and significant asymmetry in the field distributionsoccurs. Most of the field energy density flows along thesurface. Disturbances above the structure are small and theflow over the grating is not very much disturbed. In thiscase, the change in field energy directivity is imposed by thevertical shift of dielectric media, not by the horizontal mutualshift (dx) of metallic stripes, as was shown in the previoussubsection.

An interesting comparison can be made when onestructure is filled with different dielectric media in bothperiodic layers and the other structure with the same fillingsin both periodic layers. The presence of the dielectric ofthe higher refractive index changes radically the direction of

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

(a) (b)

(c) (d)

Figure 6. Two symmetrical grating configurations (dx = 0.5�) with n4 = 1, n5 = 1.5 (upper subplots (a) and (b)) and n4 = n5 = 1.5 (lowersubplots (c) and (d)), and the component of the magnetic field Hy (left column (a) and (c)) and the Poynting vector magnitude |S| with thedirection of the energy flow denoted by arrows (right column (b) and (d)).

the energy flow in comparison with the structure completelyfilled with the same dielectric medium, certainly in spite of thesubstrate. In the metallic structure exposed to dielectric withn4 = n5 = n1 = 1 the energy propagation direction is thesame as in case of a slanted metallic grating placed on a metalsubstrate [5]. However, by adding a higher refractive indexmaterial between the metal stripes, the sign of the Poyntingvector changes, and the energy flow becomes oppositelydirected. The difference in those cases, as in the previouscases, is connected with the change in the sign of the Hy

component.It seems that the difference between the refractive indices

in the grating layers n4 and n5 is crucial for obtaining the onefrom the two opposite directions of the energy flow in the two,nearly the same, configurations. They differ only slightly indx . The analysis of the Ex and Hy components shows that thefield is tightly confined in the vicinity of the metal stripes andtherefore is stronger in the cases when the stripes are embeddedin dielectrics with n4, n5 > 1, in opposition to the case of n4 =n5 = n1 = 1. The propagation constants of the SPPs movingin the opposite directions (+/− first diffraction orders) areequal in the case of normal incidence, despite the asymmetryof the grating. It appears that decisive in the unidirectionalpropagation of the SPPs are the different coupling strengths ofboth SPPs and, in consequence, different amplitudes of bothpropagating modes [5].

3.3. Action of the structures with horizontal symmetry

Finally, we present two grating configurations that aresymmetrical with respect to the y–z plane (figure 6), that is,

the structures specified by dx = 0.5�. The refractive indicesin the first configuration are n4 = 1, n5 = 1.5, in oppositionto the second one, n4 = n5 = 1.5. In both configurations, thecurls, in centres of which the energy flow is zero, emerge inthe distribution of the Poynting vector. These curls originatefrom the interference of incoming, reflected and diffractedfields, together with the propagating SPPs. This interferenceis destructive and decreases the magnitude of the Poyntingvector.

As the structure is symmetrical with respect to the y–z plane, no direction along the x-axis is preferred, and nohorizontal energy flow near the grating structure is observed.Moreover, the replacement of n4 in the first configuration byn1 in the second one changes the sign of the magnetic fieldcomponent Hy but still maintains the shape of the interferencepattern over the whole structure.

3.4. Additional remarks

The results presented above have been discussed in the contextof the energy flow direction of the total field. However,apparently similar or almost symmetric results shown for thePoynting vector evaluated over the really asymmetric gratingstructure may suggest different physical phenomena, whichare the origins of the field propagation in the two oppositedirections. In order to investigate the physical reasons forsuch behaviour of the asymmetric two-layered structure, weplot in addition the local reflection coefficient for one of theabove-mentioned geometries of the structure. Similar chartscould be plotted for other configurations, although, for clarity,we present here only one set of plots for one configuration.

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

(a) (b)

(c) (d)

Figure 7. Local reflection coefficient plotted versus the horizontal shift dx for the grating configuration as in figures 2 and 3 (n4 = 1, n5 =1.5), an imaginary part (a) and a magnitude (c) of Ex , together with a real part (b) and a magnitude (d) of Hy , of the local reflectioncoefficient of the +1 diffraction order (dotted line), −1 diffraction order (dashed line) and of all diffraction orders (solid line).

The plots in figure 7 are calculated for the case ofthe material and geometrical parameters related to those offigures 2 and 3 (i.e. n4 = 1, n5 = 1.5). They presentthe local reflection coefficients of the +1 diffraction order,the −1 diffraction order and all diffraction orders togetherfor the Ex and Hy components of the electromagnetic field.For this purpose, the local reflection coefficient is definedat some point above the structure as the total field reducedby the incident field and normalized to the incident field. Weplotted the local reflection coefficients just above the structure,at the points 2 nm over the middle of upper metal stripes,where the energy flow has the smoothest profile. Theseplots refer to numerical simulations illustrating the changein the Poynting vector sign in the presence of small structuremodifications.

Figure 7 indicates that the geometry of the structure reallyfavours the SPP excitation only in one direction. However,this probably does not exclude the change of the Poyntingvector sign and therefore energy propagation in the oppositedirection without the excitation of SPP. Our results indicatethe presence of two different kinds of phenomena which giveanalogous results in the context of the directivity of the energyflow. Strong discontinuities in the local reflection coefficient inthe case of horizontal shift slightly below dx = 0.4� relate theresonance conditions for unidirectional SPP excitation. Thechange in the directivity of energy flow near the value of dx =0.2� is connected with other physical phenomena, probablythe diffraction effects farther from the SPP resonance. Thisassumption should be verified by further research.

It appears that in the cases of too thin or too thick metalstripes, the energy flow redirection cannot be achieved. The

suppression of the directivity effect in the case of too thingratings may be explained by the weak influence of thin metalstripes on the incident wave and the quite high transmittivity ofthe structure. In this case, the metal stripes are not sufficientlyoptically thick and opaque. On the other hand, in the caseof too thick gratings, the influence of the lower subgratingon the incident field may be too weak to enable the efficientredirection of the energy flow.

The explanation of the numerical results is given hereon the grounds of the examples of the plane wave incidenceupon the infinite reflecting structures. Even this idealizedgeometry allows us to achieve a few interesting observations.They indicate that the change in the energy flow directionis possible by introduction of small differences either in thestructure geometry or in the values of the refractive index ofthe dielectric filling. The excitation efficiency of this energyflow redirection can be estimated in the idealized view of thisphenomenon as well, as is shown in figure 8. In this figure,the normalized (to the incident field amplitude) amplitudes ofthe optical fields reflected from and transmitted through thestructure are drawn. The differences between the left and theright field peaks are consistent with the direction of the energyflow preferred in the two cases described in figures 2 and 3and subsequently in figure 7.

Further analysis and more complete interpretation ofthese phenomena will be possible by taking into account amore realistic case of the finite geometry of the problem.This means the cases of beams with finite cross-sectionsincident on spatially finite periodic SPPs guiding structures.In such cases we will be able to take into account the

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J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 185401 A Roszkiewicz and W Nasalski

(a) (b)

Figure 8. Normalized amplitudes of the reflected (solid line) and transmitted (dashed line) optical fields measured above and below thetwo-layered structure for the shift dx = 0.2� and the energy flow in the +x direction (a) and for dx = 0.375� and the energy flow in the −xdirection (b).

finite excitation area of SPPs, precisely calculate theirexcitation efficiency, propagation length and other quantitativeparameters, specifying more realistically the problem underconsideration.

4. Conclusion

We have demonstrated that a simple two-layered structure,composed of two gratings with rectangular metal stripesembedded in dielectric media, can be used to controlelectromagnetic field behaviour in the vicinity of this structure.By a specific choice of the geometric configuration of thestructure, one may determine a desired direction of the energyflow. It appears that, by tailoring properly the horizontalasymmetry of the structure by shifting the gratings with respectto each other, it is possible to redirect the energy flow into oneof the two opposite directions. Moreover, the grating shiftsapplied for this purpose may be small. The directivity changein the energy flow direction is accompanied by a sign changein the magnetic field component Hy at a certain value of theshift dx . The same directivity change can also be obtained at aspecified vertical position of a contrast boundary between twodielectric filling media. The advantage of this configuration isthat it is easy to produce with the photolithography or focusedion beam techniques, which are not time consuming or costly.Moreover, in the case of metal stripes embedded in the twodifferent dielectrics, the field components are tightly confinedto the metal structure.

The geometry analyzed may find several applications inphotonics. It can serve as an optical device for controllingthe direction of electromagnetic energy flow. Maintainingthe advantages of normal incidence, it opens, at the sametime, the possibility of manipulation with the direction ofenergy propagation. Similar simple asymmetric gratingconfigurations with further optimization can serve as opticalconcentrators. By breaking the symmetry of the gratingand thus forming a subwavelength slit between two gratingsystems with the reversed shift, and properly adjusting thephase differences of SPPs incoming from both sides, a strong

concentration of light in the middle of the structure canbe obtained. Such configurations can serve as elements ofphotodetectors, where the radiation has to be concentratedwithin a small active area below the metallization region.

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