## Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts

Optics Express, Vol. 17, Issue 1, pp. 278-292 (2009)

http://dx.doi.org/10.1364/OE.17.000278

Acrobat PDF (704 KB)

### Abstract

Diffraction of plane waves on dielectric gratings with planar plasmonic inserts is studied with the emphasis put on the anomalous selectivity of diffraction orders. It is shown that some formally propagating orders can be suppressed within a wide frequency range. The effect of suppression is more general than the isolation effect observed earlier in zero-permittivity and (near-)zero-index slabs and sensitive to the frequency dependent peculiarities of the field distribution within the plasmonic layer. It is required that the real part of the permittivity of this layer is positive less than unity. The wideband features of the suppression effect, i.e., one-way transmission and diffraction-free reflection are demonstrated. Narrowband selectivity effects are also studied. The structures suggested can be used for extending the potential of technologies that are based on multibeam operation and field transformation.

© 2009 Optical Society of America

## 1. Introduction

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

2. 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]

*isolated*and the phase pattern in one region can be tailored, while being independent of the excitation shape in the other region. This effect is connected with the fact that the phase variation within an epsilon-near-zero material is small and completely disappears at zero permittivity.

*ε*≈ 0, exist in nature at terahertz frequencies (polar dielectrics) and at the visible and ultraviolet (noble metals). They are necessarily dispersive and their permittivities are often well described in the framework of Drude or Drude-Lorentz models. According to [6

6. 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]

*ε*<1 can be obtained within a wide frequency range. For example, periodic arrays of silver rods have been suggested in [7

7. 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]

2. 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]

8. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. **89**, 213902 (2002). [CrossRef] [PubMed]

9. M. Silverihna 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]

*ε*<1, e.g., see [10

10. R.W. Ziolkowski and C.-Y. Cheng, “Lumped element models of double negative metamaterial-based transmission lines,” Radio Sci. **39**, RS2017 (2004). [CrossRef]

14. W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials,” Phys. Rev. B **75**, 041102 (2007). [CrossRef]

*ε*<0, or to narrowband transmission effects at Re

*ε*>0, e.g., see [15

15. B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission through thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A **8**, S191–S207 (2006). [CrossRef]

17. M. M. Dvoynenko, I. I. Samoylenko, and J.-K. Wang, “Suppressed light transmission through corrugated metal films at normal incidence,” J. Opt. Soc. Am A **23**, 2315–2319 (2006). [CrossRef]

18. N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express **11**, 482–490 (2003). [CrossRef] [PubMed]

19. R. A. Depine, A. Lakhtakia, and D. R. Smith, “Enhanced diffraction by a rectangular grating made of a negative phase-velocity (or negative index) material,” Phys. Lett. A **337**, 155–160 (2005). [CrossRef]

20. R. A. Depine, M. E. Inchaussandague, and A. Lakhtakia, “Vector theory of diffraction by gratings made of a uniaxial dielectric-magnetic material exhibiting negative refraction,” J. Opt. Soc. Am. B **23**, 514–528 (2006). [CrossRef]

*ε*varying in line with the Drude model, i.e., the (near-)zero phase variation is not a necessary condition. For the structures considered, this leads to the appearance of several anomalous diffraction effects, which manifest themselves in the wideband suppression of some propagating diffraction orders and, hence, can be characterized in terms of the diffraction order selectivity. In particular, we will demonstrate that the one-way transmission can be achieved within a wide frequency range. It is distinguished from the recently suggested performances, in which this effect is narrowband. On the other hand, it is distinguished from a transmission that is associated with surface plasmon polaritons at negative Re

*ε*, which is a rather narrowband effect. In fact, the used plasmonic materials can be the same as those in [15

15. B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission through thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A **8**, S191–S207 (2006). [CrossRef]

18. N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express **11**, 482–490 (2003). [CrossRef] [PubMed]

3. 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]

18. N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express **11**, 482–490 (2003). [CrossRef] [PubMed]

21. 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]

23. 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]

## 2. Theoretical background

*h*is the total thickness. We refer to the structure shown in Fig. 1 as a DU structure (Double corrugations, Upper interface determines the period of the whole structure). In this case,

*M*

_{2}/

*M*

_{1}is an integer larger than unity. Correspondingly,

*M*

_{1}/

*M*

_{2}is an integer larger than unity for a DL structure (Double corrugations, Lower Interface). If

*D*=0, then we obtain a U structure. In turn, L structures correspond to

*B*=0. The upper and lower dielectric layers are assumed to be made of dielectrics with

*ε*=

*ε*

_{U}and

*ε*=

*ε*

_{L}, respectively. The permittivity of the plasmonic insert (

*a*<

*y*<

*b*) satisfies the Drude model, i.e., it is given by

*ε*(

_{i}*ω*) = 1−

*ω*

^{2}

_{p}/[

*ω*(

*ω*+

*iγ*)], where

*ω*and

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

*A*+

*B*≤ 1,

*C*−

*D*≥ 0,

*C*+

*D*≤

*a*/

*h*, and

*A*−

*B*≥

*b*/

*h*.

*y*>

*h*) and lower (

*y*<0) half-spaces are presented as follows:

*β*= [

_{n}*k*

^{2}-

*a*

^{2}

_{n}], Im

*α*=

_{n}*α*

_{0}+ 2

*πnζ*/

*L*,

*ζ*= min{

*M*

_{1},

*M*

_{2}},

*α*

_{0}=

*k*sin

*θ*,

*k*=

*ω*/

*c*is free-space wave number,

*θ*is angle of incidence,

*L*is fundamental grating period,

*ρ*and

_{n}*τ*are amplitudes of the

_{n}*n*th-order reflected and transmitted beams (Bragg waves). In turn, the incident wave is given by

*E*(

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

*y*) =

*E*

_{0}exp(

*i*

*α*

_{0}

*x*−

*i*

*β*

_{0}

*y*), where

*β*

_{0}=

*k*cos

*θ*. The intensities of the

*n*th-order reflected and transmitted beams (diffraction efficiencies) are given by

*W*means the energy of the incident wave and asterisk means complex conjugate. On the other hand,

*W*=

*R*+

*T*+

*A*where

*R*,

*T*, and

*A*are reflectance, transmittance, and absorptance, respectively.

*n*th-order beams are propagating if

*k*>

*k*where

_{n}*k*= 2

_{n}*π*∣

*n*∣/(1±∣sin

*θ*∣)

*L*. In the latter formula, signs + and − correspond to sgn

*n*≠ sgn

*θ*and sgn

*n*= sgn

*θ*, respectively. The

*n*th-order reflected beams diffract at the angles

*ψ*, which can be obtained from the equation sin

_{n}*ψ*= sin

_{n}*θ*+ 2

*πnζ*/

*kL*. It is assumed that the angles

*ψ*and

_{n}*θ*are measured from the positive-value part of the

*y*axis in the clockwise and counter-clockwise directions, respectively. The angles of diffraction of the

*n*th-order transmitted beams

*are measured from the negative-value part of the*ψ ^

_{n}*y*axis in the counter-clockwise direction, i.e.,

*=*ψ ^

_{n}*ψ*.

_{n}*t*

^{L}

_{0}=

*t*

^{U}

_{0}and

*t*

^{DL}

_{0}=

*t*

^{DU}

_{0}are satisfied for any

*kL*, provided that

*ϕ*and

_{1}*ϕ*in Eq. (1) are chosen in such a manner that a pair of L and U structures, or a pair of DL and DU structures, corresponds in fact to the same structure but the directions of illumination are opposite. These conditions are rather important for our study. Similar conditions for

_{2}*r*

^{L}

_{0}and

*r*

^{U}

_{0}, and

*r*

^{DL}

_{0}and

*r*

^{DU}

_{0}exist only if

*k*<

*k*

_{±1}. These conditions are unimportant for our purposes, since at least the ±1 st and ±2 nd orders should be propagating in order to illustrate the basic effects expected. Throughout this paper, we take

*ζ*=1 and

*ω*/

_{p}L*c*≥

*k*

_{±2}

*L*∣

_{θ=0}= 4

*π*. In case of

*ω*/

_{p}L*c*= 4

*π*and

*θ*= 0, which corresponds to most of the presented numerical results, the ±2 nd orders become propagating and Re

*ε*passes through zero nearly simultaneously, while

_{i}*kL*is raising. In Fig. 2, the dependences of

*ψ*on

_{n}*kL*and those of Re

*ε*, Im

_{i}*ε*, and ∣

_{i}*ε*∣ on

_{i}*kL*are shown, which correspond to the considered range of variation of the problem parameters.

## 3. Numerical results and discussion

### 3.1 Basic effects

*t*and

_{n}*r*vs

_{n}*kL*in the cases U and L, which correspond to the same structure being illuminated from the opposite sides. In Figs. 3(a) and 3(b), it is illuminated from the side of corrugated interface (U case). In Figs. 3(d) and 3(e), illumination is performed from the side of the flat dielectric layer (L case). One of the main effects is demonstrated in Fig. 3(a). It manifests itself in that the ±2 nd-order transmitted beams are suppressed within a wide

*kL*-range (4

*π*<

*kL*<17.2), where they are formally propagating. Therefore, the free-space wave number, starting from which the ±

*n*th-order beams actually contribute to

*T*,

*k̃*

^{T}

_{±n}, in case of

*n*= ±2 is substantially larger than

*k*

_{±2}and

*ω*/

_{p}*c*. At the same time,

*k̃*

^{T}

_{±1}and

*k̃*

^{T}

_{0}are located in the vicinity of

*k*=

*ω*/

_{p}*c*, so that

*t*

_{0}and

*t*

_{±1}contribute to

*T*within most of the range 0< Re

*ε*<1. The corrugations affect the transmitted far field, but the actual periodicity cannot be recognized by using the transmission results. From the comparison with Fig. 2(b), it is clearly seen that the range with

_{i}*T*=

*t*

_{0}+

*t*

_{-1}+

*t*

_{1}corresponds to 0< Re

*ε*<0.5. One can refer to this effect as the partial translation, or the partial isolation.

_{i}*ε*materials, where higher propagating orders in the transmission were suppressed, while the 0th order remained strongly contributive [3

3. 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]

*ε*>0 and

_{i}*t*

_{±1}≠ 0, this effect is even closer to that observed in two-dimensional dielectric and metallic photonic crystals with corrugated interfaces, where one or several propagating orders have been suppressed within a wide range of

*kL*-variation that is adjacent to an edge of a band gap [5

5. 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]

*selectivity*with respect to diffraction orders. In Fig. 3(b), one can see that all the propagating orders contribute to

*R*at 4π<

*kL*<17.2. In particular,

*k̃*

^{R}

_{±2}=

*k*

_{±2}and, therefore,

*k̃*

^{R}

_{±2}≠

*k̃*

^{T}

_{±2}. Owing to the narrowband diffraction effects,

*R*≈

*r*

_{0}at

*kL*=11.9 and

*kL*=14.8,

*T*=

*t*

_{-1}+

*t*

_{1}at

*kL*= 14.88 and

*kL*= 16.45, and

*T*=

*t*

_{0}at

*kL*=15 and

*kL*=16.55.

*t*

_{±2}≠ 0 at 4

*π*<

*kL*<17.2. This occurs due to the effect of the lower (corrugated) interface. The strong difference in the values of

*t*

^{L}

_{±2}and

*t*

^{U}

_{±2}results in the one-way transmission effect, which is similar to that observed in metallic gratings with the multiple branched slits and different periods at different interfaces [4

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

*kL*=16.45 the situation occurs when

*T*

^{U}≈

*t*

_{-1}+

*t*

_{1}≈ 0.41 and

*T*

^{L}≈

*t*

_{-2}+

*t*

_{2}≈ 0.77. Hence, all beams, that actually contribute to the transmission, can show different values of

*ψ*

_{±n}, depending on the side of incidence (

*ψ*

_{±1}= ±22.5° and

*ψ*

_{±2}= ±49.8°). The presence of the contributing ±2nd-order transmitted beams enables a substantial increase of

*T*in the L case at

*ω*/

_{p}L*c*<

*kL*<15. For example,

*T*

^{L}=0.63 and

*T*

^{U}=0.08 at

*kL*=14, where Re

*ε*≈ 0.2. It is seen in Fig. 3(e), that the corrugations do not affect the reflected far field, except for the narrow ranges with

_{i}*t*

_{±1}< 0.025 arising in the vicinity of

*kL*=16.5 and

*kL*=17.7. As a result, the isolation of two regions, i.e.,

*y*/

*h*>0.6 and

*y*/

*h*<0.4, from each other takes place, manifesting itself in that the topological features of the lower corrugated layer are not translated through the plasmonic layer. This effect is, in fact, a counterpart of the isolation effect, which has been studied in [2

2. 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]

*k̃*

^{R}

_{±2}≠

*k̃*

^{T}

_{±2}, but now also

*k̃*

^{R}

_{±1}≠

*k̃*

^{T}

_{±1}. It is noteworthy that the L structure can be reflection-free. For example,

*W*=

*T*+

*A*at

*kL*= 15.

*E*∣) within a single grating period, i.e., at 0<

_{z}*x*<

*L*and 0<

*y*<

*h*. Several examples are shown in Fig. 5, which correspond to the typical behaviors of

*t*and

_{n}*r*in Figs. 3 and 4. First, we compare Figs. 5(a) and 5(b) corresponding to the two cases (L and DL), which only differ in the shape of the upper-side interface. Three typical regions can be recognized in each figure. Within the lower region (0<

_{n}*y*/

*h*<0.4), the field patterns are nearly the same. Four maxima of ∣

*E*(

_{z}*x*,

*y*)∣ occur at

*y*=0 within the period, which can be associated with the ±2 nd orders. Indeed,

*T*≈

*t*

_{-2}+

*t*

_{2}at

*kL*=16.45 in Figs. 3(d) and 4(d). In the upper region (0.6<

*y*/

*h*<1), ∣

*E*(

_{z}*x*,

*y*)∣ in Figs. 5(a) and 5(b) is completely different, that indicates the effect exerted by the upper interface. Four maxima occur at

*y*=const at least within a part of the upper region in Fig. 5(b). In this case,

*R*≈

*r*

_{0}+

*r*

_{-2}+

*r*

_{2}, see Fig. 4(e). Since the field topology at

*y*<0.4 is not affected by the shape of the upper layer, the middle region (0.4<

*y*/

*h*<0.6) isolates the two others.

*t*

_{0}≠ 0 but the ±1 st orders dominate in

*T*and the 0th order dominates in

*R*. It is seen that the field topologies differ only within the lower region due to the shape of the lower interface. Again, the plasmonic layer reduces the periodicity, while the role of the upper and lower regions depends on the presence of corrugations. From the presented results, it follows that the structures suggested can provide one with a powerful tool for controlling the number of beams actually contributing to the transmission and reflection and the topology of the near field. One can use the following condition for U-case transmission. If

*n*∣=0,1,…

*p*, can be translated to the lower half-space (partial translation), while those corresponding to ∣

*n*∣=

*p*+1,

*p*+2,… cannot be translated (partial isolation). If the condition

*k*<

*k*

_{±1}/(Re

*ε*)

_{i}^{1/2}is satisfied, i.e.,

*p*=0 and

*k*

_{0}=0 in Eq. (5), no periodic feature can appear within the plasmonic layer. Therefore, the effect of the upper corrugated interface on the transmitted far field would be reduced, so that the condition Re

*ε*=0 is not necessary for achieving the

_{i}*total isolation*(

*T*

^{U}≈

*t*

_{0}). This case corresponds to the wave front flattening [1

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

**75**, 155410 (2007). [CrossRef]

*ε*=0.

_{i}*kL*range always exists near any

*ω*=

*ω*that all the higher orders are evanescent within the plasmonic layer, provided that 0<Re

_{p}*ε*<1 and Im

_{i}*ε*=0. Indeed,

_{i}*k*

_{±n}/(Re

*ε*)

_{i}^{1/2}= ∞ at Re

*ε*=0 and is decreased with raising Re

_{i}*ε*tending to

_{i}*k*

_{±n}at Re

*ε*=1. All the higher orders (∣

_{i}*n*∣>0) are evanescent if

*total isolation*is always a wideband effect, because the all materials with Re

*ε*<1 are necessarily dispersive. The band width is determined by the values of

_{i}*d*Re

*ε*/

_{i}*dω*and can be relatively large for the typical parameters of the Drude model. In particular, this explains why

*k*

_{±1}<

*ω*/

_{p}*c*<

*k̂*

^{T}

_{±1}. Correspondingly, the range of achievable values of the ±1 st-order diffraction angles is limited by

### 3.2 Parametric Study

*r*

_{0}≈ 0 from Fig. 6(b). The field topology within the upper region is nearly the same as in the DL case in Fig. 5(b). This does not remain true with respect to the middle and lower regions. Contrary to Fig. 5, the field at the middle region tends to lose the periodic features. Here, the behavior of ∣

*E*∣ is completely consistent with Eq. (5). Indeed,

_{z}*k*

_{±1}

*L*/(Re

*ε*)

_{i}^{1/2}=24.5 at

*kL*=16.25, where Re

*ε*=0.066 if

_{i}*ω*/

_{p}L*c*=5

*π*. This corresponds to

*p*=0 in Eq. (5). The dominance of the ±2 nd orders in

*T*is a pure diffraction effect, which appears due to peculiar wave processes within the lower region.

*θ*≠ 0. Figure 7 shows

*r*vs

_{n}*kL*in L and DL cases at

*θ*=

*π*/6 and the same other parameters as in the previous figures. In Fig. 7(a), higher orders are suppressed at least at

*kL*<16, where Re

*ε*<0.39. At 16<

_{i}*kL*<17,

*r*

_{±1}< 0.1 and

*r*

_{±2}< 0.02. In the vicinity of

*kL*=15.3 and at 17<

*kL*<18.5, the sum of

*r*at

_{n}*n*≠0 does not exceed 0.016. The wide range of the enhanced transmission appears at 16.8<

*kL*<18, instead of such a narrower range at

*θ*=0 in the vicinity of

*kL*=15 in Fig. 3(e). It follows from the comparison of Figs. 7(b) and 4(e) that

*r*

_{-2}at

*θ*=

*π*/6 can be larger than

*r*

_{-2}+

*r*

_{2}at

*θ*=0, so that the common effect of the orders with ∣

*n*∣= ±2 can be enhanced within some

*kL*-ranges, e.g., at 4

*π*<

*kL*<14.5. At the same time, the ±1 st orders and the −3 rd order are suppressed due to the isolation effect. The narrowband order selectivity effects in the reflection and transmission at

*θ*=

*π*/6 mostly correspond to sharper peaks than at

*θ*=0. In particular,

*R*≈

*r*

_{-2}> 0.9 at such a peak as seen in Fig. 7(b) at

*kL*=13.22, where

*ψ*

_{-2}=−26.8°. In this case, most of the energy of the incident wave is reflected in the near-backward direction. Note that the analogs of Eq. (5) and Eq. (6) can be derived for

*θ*≠0.

*b*/

*h*-

*a*/

*h*=0.4) and the corrugations with period

*L*are less deep. In Fig. 8, typical dependences of

*t*and

_{n}*r*on

_{n}*kL*and near-field patterns are shown. In Fig. 8(a), attention should be paid for the two effects: (i) strong wideband weakening of the propagating orders with

*n*= 0 and

*n*= ±2 at 15.7<

*kL*<17.5 and in the vicinity of

*kL*= 18, and (ii) suppression or weakening of all propagating orders at

*ω*/

_{p}L*c*<

*kL*<13.8.

*T*>0.2, which appears at

*kL*= 11.08 in the surface-plasmon regime (Res

*ε*≈ −0.29). For larger thicknesses of the dielectric layers, this effect has been observed in DU and DL cases only (compare with Figs. 3 and 4). In the comparison with Figs. 3(a) and 3 (d), both narrowband and wideband one-way transmission effects become weaker. No principal difference appears in the reflected far field, see Fig. 9(c).

_{i}*kL*-dependences of

*t*and

_{n}*r*are presented for the structures, which differ from those in Fig. 9 in that they have a thicker plasmonic layer and less deep corrugations with period

_{n}*L*. As was expected, the suppression of the ±1 st orders at

*kL*<14 in the U case in Fig. 10(a) is more pronounced than Fig. 9(a). A comparison with [5

5. 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]

*t*

_{0}than in Fig. 9. The ±2 nd orders are suppressed even at those

*kL*values, at which they contribute to

*T*in the U case in Figs. 9(a) and 3(a). Furthermore, the wideband weakening of the contribution of

*t*

_{±2}to

*T*occurs even in L case [compare Figs. 9(b) and 10(b)]. These examples show that the effect of the corrugations in producing the higher transmitted orders is weakening with a decreasing corrugation depth in both the U and L cases. The contribution of the higher orders to

*R*also becomes weaker [compare Figs. 9(c) and 10(c)].

*t*and

_{n}*r*vs

_{n}*kL*and near-field patterns are presented for the structures, which differ from those in Fig. 4 in that the dielectric layers with the corrugations of period

*L*are simply removed now. In fact, these structures show period

*L*/2. However, for the sake of convenience, the same notations are kept as above, i.e., we assume here that

*t*2

_{n+1}≡ 0 . In Fig. 11(a), all the propagating orders contribute to

*T*. Figure 11(b) demonstrates the wideband suppression of

*r*

_{±2}due to the isolation effect. A comparison of Figs. 11(b) and 3(e) shows that the absence of the upper dielectric layer and the changing of the periodicity of the whole structure do not affect the appearance of this effect.

*kL*=16 in Figs. 11(a) and 11(b). Since there are no upper-side corrugations and

*k*

_{±2}

*L*/(Re

*ε*)

_{i}^{1/2}=20.3 where Re

*ε*=0.383 at

_{i}*kL*=16, the ±2 nd orders do not contribute to the field within most part of the plasmonic layer, as well as above it. At the same time, the lower dielectric layer is thick enough in order to transform the field features seen within the plasmonic layer, which are associated with the 0th order, to those of the ±2 nd orders at

*y*=0.

*t*

_{2}<10

^{-3}at least at

*kL*<16. The strength of suppression can be varied due to a proper choice of the grating parameters. The solid and dashed lines in Fig. 12 almost coincide, so that the used number of discretization points over

*x*and

*y*,

*P*and

*Q*[23

23. 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]

*Ω*is presented in Fig. 12, showing that

*t*

_{2}>Ω can be obtained by increasing

*P*and

*Ω*. It is important that a rather good convergence of

*t*

_{2}is achieved even if

*Q*can further be decreased by appropriately increasing

*P*and

*Q*. It is noteworthy that at least

*P*=300 and

*Q*=600 were used in all examples presented in Figs. 2–11.

## 4. Conclusions

*diffraction order selectivity*, are affected by two-side corrugations, the thicknesses of plasmonic and dielectric layers, and the presence of a non-corrugated dielectric layer. The isolation effect can result in the fact that the actual cutoff wave numbers are shifted towards larger values compared to those corresponding to Rayleigh frequencies. Furthermore, the actual cut offs for the same orders in reflection and transmission can be different, that is distinguished from conventional diffraction gratings. This effect is similar to that observed earlier in photonic crystals with one-side grating-like corrugations. However, one-way transmission can be achieved in the considered structures within wider frequency ranges. As a next step, the possibility of obtaining selectivity at enhanced (nearly total) transmission will be studied. The demonstrated effects promise to be a basis for a new approach to efficient beam management in multibeam regime and field transformations.

## Acknowledgments

## References and links

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

2. | 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 |

3. | 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 |

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

5. | 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 |

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

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

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

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

10. | R.W. Ziolkowski and C.-Y. Cheng, “Lumped element models of double negative metamaterial-based transmission lines,” Radio Sci. |

11. | N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, Th. Koschny, and C. M. Soukoulis, “Magnetic response of split-ring resonators in the far-infrared frequency regime,” Opt. Lett. |

12. | L. Jylhä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. |

13. | K. Guven, A. O. Cakmak, M. D. Caliskan, T. F. Gundogdu, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Bilayer metamaterial: analysis of left-handed transmission and retrieval of effective medium parameters,” J. Opt. A |

14. | W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials,” Phys. Rev. B |

15. | B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission through thin metal film with periodically and randomly corrugated surfaces,” J. Opt. A |

16. | I. R. Hooper and J. R. Sambles, “Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces,” Phys. Rev. B |

17. | M. M. Dvoynenko, I. I. Samoylenko, and J.-K. Wang, “Suppressed light transmission through corrugated metal films at normal incidence,” J. Opt. Soc. Am A |

18. | N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express |

19. | R. A. Depine, A. Lakhtakia, and D. R. Smith, “Enhanced diffraction by a rectangular grating made of a negative phase-velocity (or negative index) material,” Phys. Lett. A |

20. | R. A. Depine, M. E. Inchaussandague, and A. Lakhtakia, “Vector theory of diffraction by gratings made of a uniaxial dielectric-magnetic material exhibiting negative refraction,” J. Opt. Soc. Am. B |

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

22. | 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). |

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

24. | R. Petit, Ed., |

**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: October 13, 2008

Revised Manuscript: November 17, 2008

Manuscript Accepted: November 21, 2008

Published: January 2, 2009

**Citation**

A. E. Serebryannikov and Ekmel Ozbay, "Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts," Opt. Express **17**, 278-292 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-1-278

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### References

- R.W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004). [CrossRef]
- 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]
- 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]
- 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, 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]
- 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. Silverihna 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]
- R. W. Ziolkowski and C.-Y. Cheng, "Lumped element models of double negative metamaterial-based transmission lines," Radio Sci. 39, RS2017 (2004). [CrossRef]
- N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, Th. Koschny, and C. M. Soukoulis, "Magnetic response of split-ring resonators in the far-infrared frequency regime," Opt. Lett. 30, 1348-1350 (2005). [CrossRef] [PubMed]
- L. Jylhä, I. Kolmakov, S. Maslovski, and S. Tretyakov, "Modeling of isotropic backward-wave materials composed of resonant spheres," J. Appl. Phys. 99, 043102 (2006). [CrossRef]
- K. Guven, A. O. Cakmak, M. D. Caliskan, T. F. Gundogdu, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, "Bilayer metamaterial: analysis of left-handed transmission and retrieval of effective medium parameters," J. Opt. A 9, S361-S365 (2007). [CrossRef]
- W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, "Electrically resonant terahertz metamaterials," Phys. Rev. B 75, 041102 (2007). [CrossRef]
- B. Baumeier, T. A. Leskova, and A. A. Maradudin, "Transmission through thin metal film with periodically and randomly corrugated surfaces," J. Opt. A 8, S191-S207 (2006). [CrossRef]
- I. R. Hooper and J. R. Sambles, "Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces," Phys. Rev. B 70, 045421 (2004). [CrossRef]
- M. M. Dvoynenko, I. I. Samoylenko, and J.-K. Wang, "Suppressed light transmission through corrugated metal films at normal incidence," J. Opt. Soc. Am A 23, 2315-2319 (2006). [CrossRef]
- N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, "Resonant optical transmission through thin metallic films with and without holes," Opt. Express 11, 482-490 (2003). [CrossRef] [PubMed]
- R. A. Depine, A. Lakhtakia, and D. R. Smith, "Enhanced diffraction by a rectangular grating made of a negative phase-velocity (or negative index) material," Phys. Lett. A 337, 155-160 (2005). [CrossRef]
- R. A. Depine, M. E. Inchaussandague, and A. Lakhtakia, "Vector theory of diffraction by gratings made of a uniaxial dielectric-magnetic material exhibiting negative refraction," J. Opt. Soc. Am. B 23, 514-528 (2006). [CrossRef]
- 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).
- 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]
- R. Petit, Ed., Electromagnetic theory of gratings (Springer, Berlin Heidelberg New York, 1980).

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