## Unidirectional transmission in non-symmetric gratings containing metallic layers

Optics Express, Vol. 17, Issue 16, pp. 13335-13345 (2009)

http://dx.doi.org/10.1364/OE.17.013335

Acrobat PDF (1332 KB)

### Abstract

The mechanism of achieving unidirectional transmission in the gratings, which only contain isotropic dielectric and metallic layers, is suggested and numerically validated. It is shown that significant transmission in one direction and nearly zero transmission in the opposite direction can be obtained in the same intrinsically isotropic gratings as those studied recently in A. E. Serebryannikov and E. Ozbay, Opt. Express **17**, 278 (2009), but at a non-zero angle of incidence. The tilting, non-symmetric features of the grating and the presence of a metallic layer with a small positive real part of the index of refraction are the conditions that are necessary for obtaining the unidirectionality. Single- and multibeam operational regimes are demonstrated. The frequency and angle ranges of the unidirectional transmission can be estimated by using the conventional framework based on isofrequency dispersion contours and construction lines that properly take into account the periodic features of the interfaces, but should then be corrected because of the tunneling arising within the adjacent ranges. After proper optimization, this mechanism is expected to become an alternative to that based on the use of anisotropic materials.

© 2009 Optical Society of America

## 1. Introduction

1. Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in gyromagnetic photonic crystals,” Phys. Rev. Lett. **100**, 013905 (2008). [CrossRef] [PubMed]

3. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals,” J. Magnetism Magnet. Mat. **300**, 117 (2006). [CrossRef]

4. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B **67**, 165210 (2003). [CrossRef]

5. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B **65**, 201104 (2002). [CrossRef]

6. A. E. Serebryannikov, T. Magath, and K. Schuenemann, “Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface,” Phys. Rev. E **74**, 066607 (2006). [CrossRef]

7. M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E **74**, 056611 (2006). [CrossRef]

8. A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, “Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials,” Phys. Rev. B **73**, 115111 (2006). [CrossRef]

*partial unidirectionality*. The different periods of the two interfaces, i.e. no symmetry with respect to the middle plane, is a common feature of the mentioned three classes of the periodic structures.

9. A.E. Serebryannikov and E. Ozbay, “Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts,” Opt. Express **17**, 278 (2009). [CrossRef] [PubMed]

*isolation*, because the far field in the transmission half-space does not contain any order that could be created owing to the corrugation of the front-side (illumination) interface, or it contains only a part of these orders. This effect is similar to the isolation that was studied recently for the slabs made of epsilon-near-zero [10

10. A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B **75**, 155410 (2007). [CrossRef]

11. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E **70**, 046608 (2004). [CrossRef]

9. A.E. Serebryannikov and E. Ozbay, “Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts,” Opt. Express **17**, 278 (2009). [CrossRef] [PubMed]

12. J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, “Metallodielectric gratings with subwavelength slots: Optical properties,” Phys. Rev. B **68**, 205103 (2003). [CrossRef]

15. D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, “Multiple wavelength filtering of light through inner resonances,” Phys. Rev. E **72**, 066610 (2005). [CrossRef]

*imperfect unidirectionality*can be obtained for the gratings with layers showing

*isotropic*dispersion, so that significant transmission occurs if the grating is illuminated from the side of the corrugated interface but vanishes if it is illuminated from the side of the flat interface. Hence, an anisotropy-like transmission can be obtained even without anisotropy-like dispersion. This can be achieved due to tilting, which makes the zero order in air and dielectric layer(s) uncoupled to that in the collisionless metallic layer, while at least one higher order is coupled. Furthermore, the dominance of certain orders in transmission can be realized by varying the angle of incidence and choosing proper parameters of the corrugations and metallic layer. In particular, single-beam and multibeam unidirectionality with the equal contribution of higher orders, or dominant contribution of one of them will be demonstrated.

## 2. Theoretical background

*B*=0 and

*D*≠0, we obtain an L structure. In turn, U structures correspond to

*B*≠0 and

*D*=0.

*ε*

_{U}and

*ε*

_{L}stand for the permittivity of the upper and lower dielectric layers, respectively. The permittivity of the metallic layer (

*a*<

*y*<

*b*) varies with the angular frequency

*ω*according to the Drude model:

*ω*and

_{p}*γ*mean the plasma and collision frequencies. It is assumed that

*A*+

*B*≤1,

*C*−

*D*≥0,

*C*+

*D*≤

*a*,

*A*−

*B*≥

*b*, so that for the maximal thickness of the grating,

*h*, we have

*h*=

*A*+

*B*−(

*C*−

*D*). Consideration is restricted to the case of

*s*-polarization. The incident wave is given by

*E*(

^{i}_{z}*x*,

*y*)=

*E*

_{0}exp(

*iα*

_{0}

*−*

^{x}*iβ*

_{0}

*y*), where

*α*

_{0}=

*k*sin

*θ*,

*β*

_{0}=

*k*cos

*θ*,

*k*=

*ω*/

*c*, and

*θ*is the angle of incidence. The diffraction efficiencies are given by

*W*is the energy of the incident wave,

*τ*and

_{n}*ρ*are the nth-order transmission and reflection coefficients,

_{n}*β*=[

_{n}*k*

^{2}−

*α*

^{2}

*]*

_{n}^{1/2},

*α*=

_{n}*α*

_{0}+2

*πM*/

*L*and the asterisk means complex conjugate.

*W*=

*R*+

*T*+

*A*where

*R*,

*T*, and

*A*are reflectance, transmittance, and absorptance, respectively. It is expected that the order selectivity can be realized due to the effect of the metallic layer, while the dielectric layers should provide a high index contrast at the interfaces. Hence, the contribution of

*t*and

_{n}*r*to

_{n}*T*and

*R*can be enhanced or weakened due to the proper choice of

*ε*

_{U}and

*ε*

_{L}. The nth-order beam propagates in air at

*k*>|

*α*

_{0}+2

*πnM*/

*kL*|. The angles of diffraction

*ϕ*depend on

_{n}*θ*according to the equation

*ϕ*=sin

_{n}*θ*+2

*πM*/

*kL*.

*n*th order propagates in a collisionless Drude medium (

*γ*=0) starting from a larger

*k*than in air, i.e.

*k*=

*κ*>|

_{n}*α*

_{0}+2

*πnM*/

*kL*|, which can be found at

*M*=1 from the following equation:

*χ*=cos

*θ*. At

*n*=0,

*κ*

_{0}

*L*=

*ω*

_{p}*L*/

*cχ*. Correspondingly, the range of the variation of

*θ*, in which the propagating

*n*th order in air or dielectric is coupled to the propagating nth order in a Drude medium at fixed

*kL*, is limited by

*ψ*<

_{n}*θ*<

*φ*, where

_{n}*n*>0, and

*n*<0, provided that Ω

*/*

_{n}*kL*<−1 in Eq. (4) and Θ

*/*

_{n}*kL*>1 in Eq. (5). In turn, Θ

*=−2*

_{n}*πn*+Λ

*and Ω*

_{n}*=−2*

_{n}*πn*−Λ

*, and*

_{n}*ω*-domain thresholds,

*κ*, and

_{n}*θ*-domain boundaries of the

*n*th-order propagation range,

*ψ*and

_{n}*φ*, can be visualized by using the conventional approach based on the isofrequency dispersion contours for the dielectric and metallic media and the construction lines, which take into account the periodic features of the interfaces and the orientation of the corrugated interface with respect to the incident wave.

_{n}*k*=

*κ*

_{±1}and

*θ*=0, so that zero order is only coupled to a propagating order in a collisionless metal, regardless of whether the corrugated interface (U case) or non-corrugated interface (L case) is illuminated. As a result, a single transmission channel is open in both cases and transmission is reciprocal, i.e.

*t*

^{L}

_{0}=

*t*

^{U}

_{0}. This case corresponds to the

*bidirectional isolation*regime, since no corrugation feature is transferred through the structure in the U case. In the L case, negative-first-order transmission is expected to occur due to the effect of the corrugated (back-side) interface, leading to the fact that

*t*

^{L}

*≠*

_{n}*t*

^{U}

*for the propagating orders with |*

_{n}*n*|=1 and, hence, to one-way transmission like that studied in [9

9. A.E. Serebryannikov and E. Ozbay, “Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts,” Opt. Express **17**, 278 (2009). [CrossRef] [PubMed]

*T*

^{U}and zero

*T*

^{L}can be realized at least if the propagating zero order in the corrugated dielectric layer is uncoupled to the zero order in metal at

*γ*=0, while at least one of the higher orders remains coupled. The right subplot in Fig. 1(b) demonstrates that this can simply be obtained by tilting. Here,

*k*=

*κ*

_{−2}=

*κ*

_{0}so that the order with

*n*=−1 is only coupled in the U case. At the same time, no transmission channel is open in the L case because of the absence of corrugations on the front-side interface.

*ω*than the zero-order one, i.e. the condition

*n*|>0. Calculations of

*κ*vs

_{n}L*θ*and

*ω*were performed using Eq. (3). The obtained results are shown in the left plot in Fig. 2. It is seen that

_{p}*κ*

_{0}and

*κ*at |

_{n}*n*|>0 may show the opposite trends of variation with

*θ*. Based on these results, the zones of min

*κ*have been detected, which correspond to different

_{n}L*n*, see the right plot in Fig. 2. The dashed line corresponds to min

*κ*=

_{n}L*κ*

_{0}

*L*that is the boundary of the zone where unidirectional transmission might appear. This line is given by the equation

*κ*

_{0}=

*κ*

_{−1}where

*κ*

_{0}and

*κ*

_{−1}are replaced by the right-hand side of Eq. (3) for the corresponding values of

*n*. The boundary between the ranges of min

*κ*=

_{n}L*κ*

_{−1}

*L*and min

*κ*=

_{n}L*κ*

_{−2}

*L*is given by

## 3. Results and discussion

### 3.1 Basic effects

**17**, 278 (2009). [CrossRef] [PubMed]

*kL*-range the unidirectional transmission is strongly pronounced, so that

*T*

^{U}/

*T*

^{L}≥100 at

*kL*≥13 and max

*T*

^{U}/

*T*

^{L}=970 at

*kL*≈16.2. This feature is in agreement with the interpretation based on the dispersion results. Indeed, when the structure is illuminated from the side of the corrugated dielectric layer (U case), the zero order in the dielectric is not coupled to a propagating order in the collisionless metal, that leads to

*t*

^{U}

_{0}≈0.

*T*

^{U}, as seen in Fig. 3(a). If the same structure is illuminated from the side of the non-corrugated inteface (L case), there is only one order (zero order), which propagates in the dielectric layer, but it is not coupled to that in the metal. This example demonstrates that an anisotropic-like unidirectional transmission can be achieved due to the tilting of the structures, which are intrinsically isotropic. The non-zero values of

*T*

^{L}can be connected with a weak tunneling.

*κ*obtained from Eq. (3), and the values of

_{n}*k*=

*K*starting from which higher orders actually contribute to

_{n}*T*

^{U}. For the parameters used in Fig. 3(a),

*κ*is equal to 8

_{n}L*π*, 13.78, 12.67, 14.18, and 16.56 for

*n*=0,-1,-2,-3, and -4, respectively, while

*κ*

_{0}

*c*/

*ω*=2. The values of

_{p}*ϕ*at

_{n}*n*=−1,-2,-3, and -4 are equal to 24.2°, -6.8°, -27.6°, and -40.7°. Comparing to Fig. 3(a), one can see that Eq. (3) correctly predicts the order of the actual thresholds at various

*n*with respect to each other. The values of

*K*, starting from which

_{n}L*T*

^{U}>10

^{−3}, are equal to 10.62, 10.18, 12.73, and 15.88 for

*n*=−1,−2,−3, and -4, respectively. The widening of the range of the actual contribution of the

*n*th order at

*k*<

*κ*takes place due to the contribution of evanescent waves. However,

_{n}*t*keep relatively small values here, while the enhancement of their contribution occurs in the vicinity of

_{n}*k*=

*κ*.

_{n}*θ*=0 in [9

**17**, 278 (2009). [CrossRef] [PubMed]

*b*−

*a*)/

*h*results in the weakening of the tunneling, especially in the L case, so that now

*T*

^{U}/

*T*

^{L}≥10

^{5}at

*kL*≥12.74 and max

*T*

^{U}/

*T*

^{L}=2.1×10

^{6}at

*kL*≈13.2. The values of

*K*are equal to 13.26, 12.44, 14.15, and 16.59 for

_{n}L*n*=−1,−2,−3, and −4, respectively. This means that the tunneling is weakened so that

*K*≈

_{n}*κ*at least for

_{n}*n*<−2. It is noteworthy that the

*single-beam unidirectional regime*in Fig. 4(a) is realized due to the order with

*n*=−2, i.e. at

*K*

_{−2}<

*k*<

*K*

_{−1}, showing the maximum of

*t*

_{−2}=0.22 at

*kL*=13.22 and

*ϕ*

_{−2}=−4.8°. In the comparison, in Fig. 3(a),

*t*

_{−2}=0.18 and

*t*

_{−1}=0.01 at

*kL*=13.47 (

*ϕ*

_{−2}=−3.8° and

*ϕ*

_{−1}=23.6°). The

*multibeam unidirectional regime*with

*t*

_{−2}=

*t*

_{−3}=0.22 is observed in Fig. 4(a) at

*kL*=17.6. The largest values of

*T*

^{U}are achieved in Fig. 4(a) at

*kL*=17.88 (

*T*

^{U}=0.64) and in Fig. 3(a) at

*kL*=18.8 (

*T*

^{U}=0.69).

### 3.2 Gratings with smaller thickness

*t*vs

_{n}*kL*in the U case for the grating with the same (

*b*–

*a*)/

*L*but smaller

*h*/

*L*than in Fig. 3. For the L-case counterpart,

*t*

^{L}

*≪*

_{n}*t*

^{U}

*at least within the ranges of the significant contribution of the several first orders to*

_{n}*T*

^{U}in a similar fashion as in Figs. 3 and 4. Therefore, we further consider the U case only.

*T*

^{U}and

*T*

^{L}can also be achieved when

*κ*

_{0}

*c*/

*ω*is smaller than in Figs. 3 and 4. At

_{p}*θ*=

*π*/6,

*κ*=14.51, 4

_{n}L*π*, 13.79, 16.45, and 19.76 for

*n*=0,-1,-2,-3, and -4, respectively, and

*κ*

_{0}

*c*/

*ω*=1.155. Hence, it is expected that the order with

_{p}*n*=−1 prevails in the propagating-wave originated transmission in the range right above

*kL*=

*ω*

_{p}*L*/

*c*. All the orders contribute to

*T*

^{U}starting from

*k*=

*K*<

_{n}*κ*due to tunneling.

_{n}*t*

_{−1}remains the basic contributor at least at

*kL*<15, while max

*t*

_{−1}≈0.3 and

*T*

^{U}≈0.32 at

*kL*=14.87. Furthermore, the negative first order dominates in the evanescent-wave originated transmission, i.e. at

*kL*<

*ω*/

_{p}L*c*, where

*t*

_{−1}reaches 0.05 at

*kL*=11.42. Within the range of large

*t*

^{U}

_{0}, the partial unidirectionality leads, in particular, to the

*unidirectional splitting*at

*kL*=16.43, where

*t*

^{U}

_{−1}=

*t*

^{U}

_{−2}≈0.14 and

*T*

^{L}=

*t*

^{L}

_{0}=

*t*

^{U}

_{0}. This regime is similar to those studied at

*θ*=0 in [9

**17**, 278 (2009). [CrossRef] [PubMed]

*kL*-range with the dominant contribution of the order with

*n*=−1 also appears at

*θ*=

*π*/3, see Fig. 5(b). In this case, the values of

*κ*are the same as in Figs. 3(a) and 4(a). However, in contrast to these figures,

_{n}L*t*

_{−2}<0.101 in Fig. 5(b) within the range of the dominant contribution of the order with

*n*=−2. On the other hand, max

*t*

^{U}

_{−1}≈0.43 and

*T*

^{U}≈0.52 at

*kL*=16.48. In fact, this case can be considered as the unidirectional transmission with one main and two parasitic beams.

*t*vs

_{n}*kL*are presented for the two thin two-layer gratings, showing in the U case the upper corrugated dielectric layer and lower metallic flat layer. In Fig. 6(a), the actual grating period is two times smaller than in the previous figures. However, the order nomenclature is kept the same for the purposes of a comparison, i.e. we assume that

*t*

_{2n+1}≡0 at

*n*=0,±1,±2,…. The lamellar grating in Fig. 6(b) is topologically similar to but distinguished in material of the flat layer and bars from the gratings considered in the studies of surface-plasmon assisted transmission through the subwavelength slits, that occurs for

*p*-polarization [12

12. J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, “Metallodielectric gratings with subwavelength slots: Optical properties,” Phys. Rev. B **68**, 205103 (2003). [CrossRef]

14. A. Barbara, P Quemerais, E. Bustarret, and T. Lopez-Rios, “Optical transmission through subwavelength metallic gratings,” Phys. Rev. B **66**, 161403 (2002). [CrossRef]

*kL*=

*K*

_{−2}

*L*<11 and

*kL*=

*K*

_{−4}

*L*=15.81. It is also noteworthy that

*K*

_{−2}

*L*is nearly the same here as in Fig. 3(a), but smaller than in Fig. 4(a). Figure 6(b) illustrates the peculiar features of

*t*vs

_{n}*kL*for the lamellar grating with the same metallic layer as in Figs. 3, 5, and 6(a). Tunneling at

*k*<

*K*appears in Fig. 6(b) in such a way that

_{n}*t*

_{−1}and

*t*

_{−2}exceed 0.01 at nearly the same value of

*kL*≈12.1, so that the range of a significant dominance of the order with

*n*=−2 does not exist. As a result, only the multibeam unidirectionality can be obtained. In Figs. 6(a) and 6(b), max

*T*

^{U}>0.6, i.e. nearly the same portion of the incident-wave energy is transmitted as for the thick gratings shown in Figs. 3(a) and 4(a).

### 3.3 Angle-domain unidirectionality

*θ*-domain. In Fig. 7, the transmission results are presented for the same grating parameters as in Fig. 4, except for

*ω*/

_{p}L*c*=2

*π*. A smaller

*ω*/

_{p}L*c*is used here in order to avoid the simultaneous contribution of multiple higher orders, which could lead to some difficulties in the interpretation of the results. Figure 7(a) shows

*t*vs

_{n}*kL*in the U case at

*θ*=

*π*/3. One can see that the frequency-domain transmission features observed in Fig. 4(a) remain. The only difference is that now the order with

*n*=−1 appears first in the vicinity of

*ω*=

*ω*, leading to the single-beam wideband unidirectionality. This is in agreement with Eq. (3). For the used parameters,

_{p}*κ*is equal to 4

_{n}L*π*, 6.33, 8.28, and 11.14 for

*n*=0,-1,-2, and -3, respectively, while

*κ*

_{0}

*c*/

*ω*=2. In addition to the single-beam wideband unidirectionality, the narrowband effect occurs at

_{p}*kL*=10 where

*t*

_{−1}=0.38 and

*T*=0.395. A similar narrowband behavior is often observed in the conventional dielectric gratings, but here it manifests itself in a unidirectional fashion due to the stacking of the dielectric and metallic layers.

*kL*-value,

*t*vs

_{n}*θ*are shown for the U case in Fig. 7(b) and for the L case in Fig. 7(c). It is seen from the comparison of Fig. 7(b) with Fig. 7(c) that the range of 50°<

*θ*<80° can be considered as that of the two-beam imperfect unidirectionality, since

*t*

^{U}

_{0}=

*t*

^{L}

_{0}<0.02 and

*t*

^{U}

_{−1}>0.16. At

*θ*>68°,

*t*

^{U}

_{0}=

*t*

^{L}

_{0}<10

^{−3}. The ranges of the significant contribution of the orders with

*n*=0, +1 and -2 to

*T*

^{U}are in good agreement with Eq. (5). In particular,

*φ*

^{U}

_{0}=51.1°,

*φ*

^{U}

_{+1}=8.6° and

*ψ*

^{U}

_{−2}=28.6°. The ranges of the actual contribution are a bit wider due to the tunneling. In turn, the values of

*φ*

^{L}

_{+1}=21.8° and

*ψ*

^{L}

_{−2}=14.9° obtained for the back-side air half-space correspond well to the boundaries of the range of the actual contribution of the orders with

*n*=+1 and

*n*=−2 to

*T*

^{L}, demonstrating the role of back-side corrugations in the appearance of non-zero

*t*at |

_{n}*n*|>0, while the higher-order transmission channels in metal are closed. The differences in the values of

*φ*and

_{n}*ψ*in the U and L cases result in the appearance of wide

_{n}*θ*-ranges, where higher orders show one-way transmission. The anomalous narrowband transmission effect occurs at

*θ*=22.25°, where

*t*

_{0}<0.01,

*T*

^{L}>0.2 and

*T*

^{U}<0.04, so that the transmission is suppressed while illuminating the grating from the side of the corrugated interface, leading to

*inverse*unidirectionality.

**17**, 278 (2009). [CrossRef] [PubMed]

*ω*-domain, which manifests itself in that the back-side corrugations in the L case do not lead to the appearance of the diffracted beams (|

*n*|>0) in the half-space above the front-side interface. Figure 8 demonstrates the similar effect arising in

*θ*-domain. According to Eq. (5),

*ψ*

^{L}

_{−1}=6.52° and

*ψ*

^{L}

_{−2}=61.1° in Fig. 8(a),

*ψ*

^{L}

_{−1}=−1.02° and

*ψ*

^{L}

_{−2}=42.9° in Fig. 8(b), and

*ψ*

^{L}

_{−1}=8.6° and

*ψ*

^{L}

_{−2}=28.6° in Fig. 8(c),

*ψ*

^{L}

*=−*

_{n}*φ*

^{L}

_{−n}. One can see that the ranges of the actual contribution of higher orders to

*R*

^{L}are much narrower. They do not contribute within any of the zero-order forbidden

*θ*-ranges, i.e. at

*θ*>40.4° in Fig. 8(a),

*θ*>45.7° in Fig. 8(b), and

*θ*>51.1° in Fig. 8(c), and within smaller-

*θ*ranges where the zero order is propagating in metal in line with Eq. (5), and hence at least one channel is open. The difference between the propagating-wave ranges with the vanishing and substantial role of the corrugations cannot be detected from the dispersion results. Despite the narrowing ranges of higher-order reflection, its contribution to

*R*

^{L}can still be significant, e.g. in the vicinity of

*θ*=20° in Fig. 8(a). The above-reported diffraction features remain within a wide range of parameter variation. As an example, Fig. 9 shows

*t*vs

_{n}*θ*for a grating, which differs from that in Fig. 3 in smaller

*ω*/

_{p}L*c*. Now, it is the same as in Figs. 7 and 8. The appearance of the multibeam unidirectionality at

*θ*>

*π*/3 can be seen from the comparison of Figs. 9(a) and 9(b). Here, max

*T*

^{U}=0.51 at

*θ*=63.5° while

*T*

^{L}tends to vanish.

*T*

^{U}can be increased due to the proper choice of the

*kL*value. For example, in Fig. 9(c),

*T*≈2

*t*

^{U}

_{−1},

*t*

^{U}

_{−1}=

^{t}

^{U}

_{−2}=0.297, and

*T*

^{L}=0.044 at θ=64.15°.

### 3.4 Unidirectionality in purely metallic gratings

*t*vs

_{n}*kL*for the two thin gratings with the same values of

*ω*/

_{p}L*c*and

*κ*, and

_{n}L*θ*as in Fig. 5(a). Correspondingly, the single-beam unidirectionality is realized due to the negative first order. Note that

*t*

_{−2}at

*k*>

*κ*

_{−2}and

*t*

_{−3}at

*k*>

*κ*

_{−3}show here rather small values. Furthermore,

*t*

_{−3}in Fig. 10(a) nearly vanishes. Therefore, the additional selectivity of the diffraction orders is expected to be realizable due to the optimization of the corrugation parameters and the use of a single metallic corrugated layer, instead of a two- or three-layer grating.

## 4. Conclusions

## Acknowledgments

## References and links

1. | Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in gyromagnetic photonic crystals,” Phys. Rev. Lett. |

2. | F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” arXiv:cond-mat/0503588 (2008). |

3. | A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals,” J. Magnetism Magnet. Mat. |

4. | A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B |

5. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B |

6. | A. E. Serebryannikov, T. Magath, and K. Schuenemann, “Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface,” Phys. Rev. E |

7. | M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E |

8. | A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, “Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials,” Phys. Rev. B |

9. | A.E. Serebryannikov and E. Ozbay, “Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts,” Opt. Express |

10. | A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B |

11. | R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E |

12. | J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, “Metallodielectric gratings with subwavelength slots: Optical properties,” Phys. Rev. B |

13. | K. G. Lee and Q-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. |

14. | A. Barbara, P Quemerais, E. Bustarret, and T. Lopez-Rios, “Optical transmission through subwavelength metallic gratings,” Phys. Rev. B |

15. | D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, “Multiple wavelength filtering of light through inner resonances,” Phys. Rev. E |

16. | A. Sihvola, |

17. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

18. | B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B |

19. | S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. |

20. | M. Silverinha and N. Engheta, “Design of matched zero-index metamaterials using non-magnetic inclusions in epsilon-near-zero media,” Phys. Rev. B |

21. | T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(050.1970) Diffraction and gratings : Diffractive optics

(120.7000) Instrumentation, measurement, and metrology : Transmission

(160.3900) Materials : Metals

(160.4670) Materials : Optical materials

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 11, 2009

Revised Manuscript: July 9, 2009

Manuscript Accepted: July 13, 2009

Published: July 20, 2009

**Citation**

A. E. Serebryannikov and Ekmel Ozbay, "Unidirectional transmission in non-symmetric gratings containing metallic layers," Opt. Express **17**, 13335-13345 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13335

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### References

- Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008). [CrossRef] [PubMed]
- F. D. M. Haldane and S. Raghu, "Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry," arXiv:cond-mat/0503588 (2008).
- A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magnetism Magnet. Mat. 300, 117 (2006). [CrossRef]
- A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003). [CrossRef]
- C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002). [CrossRef]
- A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006). [CrossRef]
- M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006). [CrossRef]
- A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006). [CrossRef]
- A.E. Serebryannikov and E. Ozbay, "Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts," Opt. Express 17, 278 (2009). [CrossRef] [PubMed]
- A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007). [CrossRef]
- R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004). [CrossRef]
- J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003). [CrossRef]
- K. G. Lee and Q-H. Park, "Coupling of surface plasmon polaritons and light in metallic nanoslits," Phys. Rev. Lett. 95, 103902 (2005). [CrossRef] [PubMed]
- A. Barbara, P, Quemerais, E. Bustarret, and T. Lopez-Rios, "Optical transmission through subwavelength metallic gratings," Phys. Rev. B 66, 161403 (2002). [CrossRef]
- D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005). [CrossRef]
- A. Sihvola, Electromagnetic mixing formulas and applications (IEE, London, 1999). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
- B. T. Schwartz and R. Piestun, "Total external reflection from metamaterials with ultralow refractive index," J. Opt. Soc. Am. B 20, 2448-2453 (2003). [CrossRef]
- S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]
- M. Silverinha and N. Engheta, "Design of matched zero-index metamaterials using non-magnetic inclusions in epsilon-near-zero media," Phys. Rev. B 75, 075119 (2007). [CrossRef]
- T. Magath and A. E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A 22, 2405-2418 (2005). [CrossRef]

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