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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3075–3088
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Asymmetric transmission of terahertz waves using polar dielectrics

Andriy E. Serebryannikov, Ekmel Ozbay, and Shunji Nojima  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 3075-3088 (2014)
http://dx.doi.org/10.1364/OE.22.003075


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Abstract

Asymmetric wave transmission is a Lorentz reciprocal phenomenon, which can appear in the structures with broken symmetry. It may enable high forward-to-backward transmittance contrast, while transmission for one of the two opposite incidence directions is blocked. In this paper, it is demonstrated that ultrawideband, high-contrast asymmetric wave transmission can be obtained at terahertz frequencies in the topologically simple, i.e., one- or two-layer nonsymmetric gratings, which are entirely or partially made of a polar dielectric working in the ultralow- ε regime inspired by phonon-photon coupling. A variety of polar dielectrics with different characteristics can be used that gives one a big freedom concerning design. Simple criteria for estimating possible usefulness of a certain polar dielectric are suggested. Contrasts exceeding 80dB can be easily achieved without a special parameter adjustment. Stacking a high- ε corrugated layer with a noncorrugated layer made of a polar dielectric, one can enhance transmission in the unidirectional regime. At large and intermediate angles of incidence, a better performance can be obtained owing to the common effect of nonsymmetric diffractions and directional selectivity, which is connected with the dispersion of the ultralow- ε material. At normal incidence, strong asymmetry in transmission may occur in the studied structures as a purely diffraction effect.

© 2014 Optical Society of America

1. Introduction

The interest to the asymmetric wave transmission has been growing over last years. This is a Lorentz reciprocal phenomenon, which manifests itself in strong difference between the forward and the backward transmission in the structures with broken spatial inversion symmetry for illumination directions that differ by 180 degrees, e.g., see [1

1. M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 74(5), 056611 (2006). [CrossRef] [PubMed]

7

7. S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “Spoof-plasmon relevant one-way collimation and multiplexing at beaming from a slit in metallic grating,” Opt. Express 20(24), 26636–26648 (2012). [CrossRef] [PubMed]

]. In spite of that the functionalities associated with nonreciprocity, e.g., optical isolation cannot be obtained in the frame of this mechanism, there are many others, which are realizable in various nonsymmetric structures made of linear, isotropic, passive materials [8

8. V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012). [CrossRef] [PubMed]

,9

9. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013). [CrossRef]

]. In particular, the achievable operation regimes include single-beam deflection, diodelike transmission, and dual- and multibeam splitting [2

2. A. E. Serebryannikov and E. Ozbay, “Unidirectional transmission in non-symmetric gratings containing metallic layers,” Opt. Express 17(16), 13335–13345 (2009). [CrossRef] [PubMed]

,3

3. A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]

,6

6. A. Cicek, M. B. Yucel, O. A. Kaya, and B. Ulug, “Refraction-based photonic crystal diode,” Opt. Lett. 37(14), 2937–2939 (2012). [CrossRef] [PubMed]

,7

7. S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “Spoof-plasmon relevant one-way collimation and multiplexing at beaming from a slit in metallic grating,” Opt. Express 20(24), 26636–26648 (2012). [CrossRef] [PubMed]

]. In fact, asymmetric wave transmission is possible due to additional transmission channels, to which the incident wave is either coupled or uncoupled, depending on the illumination side. Such a channel may be created by involving a higher diffraction order or polarization state, which differs from that of the incident wave. Generally speaking, all these regimes can be explained in terms of one-way mode conversion. Various performances of nonsymmetric structures for asymmetric wave transmission in acoustic, microwave, and optical ranges have been suggested to this time. They include photonic crystal gratings [3

3. A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]

,4

4. X.-B. Kang, W. Tan, Z.-S. Wang, Z.-G. Wang, and H. Cheng, “High-efficiency one-way transmission by one-dimensional photonic crystal with gratings on one side,” Chin. Phys. Lett. 27(7), 074204 (2010). [CrossRef]

,10

10. S. Xu, C. Qiu, and Z. Liu, “Acoustic transmission through asymmetric grating structures made of cylinders,” J. Appl. Phys. 111(9), 094505 (2012). [CrossRef]

12

12. A. E. Serebryannikov, K. B. Alici, T. Magath, A. O. Cakmak, and E. Ozbay, “Asymmetric Fabry-Perot-type transmission in photonic-crystal gratings with one-sided corrugations at a two-way coupling,” Phys. Rev. A 86(5), 053835 (2012). [CrossRef]

], stacked hole arrays with one-side corrugations [13

13. M. Beruete, A. E. Serebryannikov, V. Torres, M. Navarro-Cia, and M. Sorolla, “Toward compact millimeter-wave diode in thin stacked-hole array assisted by a dielectric grating,” Appl. Phys. Lett. 99(15), 154101 (2011). [CrossRef]

], metallic gratings with subwavelength slits [1

1. M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 74(5), 056611 (2006). [CrossRef] [PubMed]

,5

5. M. Stolarek, D. Yavorskiy, R. Kotyński, C. J. Zapata Rodríguez, J. Łusakowski, and T. Szoplik, “Asymmetric transmission of terahertz radiation through a double grating,” Opt. Lett. 38(6), 839–841 (2013). [CrossRef] [PubMed]

,7

7. S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “Spoof-plasmon relevant one-way collimation and multiplexing at beaming from a slit in metallic grating,” Opt. Express 20(24), 26636–26648 (2012). [CrossRef] [PubMed]

,14

14. M. Mutlu, S. Cakmakyapan, A. E. Serebryannikov, and E. Ozbay, “One-way reciprocal spoof surface plasmons and relevant reversible diodelike beaming,” Phys. Rev. B 87(20), 205123 (2013). [CrossRef]

], gratings that represent a one-fraction structure or a two-fraction structure obtained by either stacking two layers, one of which is corrugated, or embedding one single-fraction grating into another [2

2. A. E. Serebryannikov and E. Ozbay, “Unidirectional transmission in non-symmetric gratings containing metallic layers,” Opt. Express 17(16), 13335–13345 (2009). [CrossRef] [PubMed]

,15

15. W.-M. Ye, X.-D. Yuan, C. C. Guo, and C. Zen, “Unidirectional transmission in non-symmetric gratings made of isotropic material,” Opt. Express 18(8), 7590–7595 (2010). [CrossRef] [PubMed]

,16

16. 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(11), 115111 (2006). [CrossRef]

], and ultrathin structures based on subwavelength resonators that enable polarization manipulation [14

14. M. Mutlu, S. Cakmakyapan, A. E. Serebryannikov, and E. Ozbay, “One-way reciprocal spoof surface plasmons and relevant reversible diodelike beaming,” Phys. Rev. B 87(20), 205123 (2013). [CrossRef]

,17

17. R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009). [CrossRef]

]. Performance in terms of the asymmetry and band width can be especially strong when dispersion contributes to the resulting mechanism. Its potential has been demonstrated, for example, in photonic crystal gratings, where various regimes of wideband unidirectional transmission, i.e., those with transmission vanishing in a wide frequency range for one of the two opposite illumination directions have been obtained [3

3. A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]

,10

10. S. Xu, C. Qiu, and Z. Liu, “Acoustic transmission through asymmetric grating structures made of cylinders,” J. Appl. Phys. 111(9), 094505 (2012). [CrossRef]

]. Moreover, inclining interfaces, as in photonic crystal prisms, allows one to further extend variety of the regimes that are achievable in the frame of the asymmetric wave transmission mechanism [6

6. A. Cicek, M. B. Yucel, O. A. Kaya, and B. Ulug, “Refraction-based photonic crystal diode,” Opt. Lett. 37(14), 2937–2939 (2012). [CrossRef] [PubMed]

,18

18. C. Wang, X.-L. Zhong, and Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012). [PubMed]

,19

19. J. H. Oh, H. W. Kim, P. S. Ma, H. M. Seung, and Y. Y. Kim, “Inverted bi-prism phononic crystals for one-sided elastic wave transmission applications,” Appl. Phys. Lett. 100(21), 213503 (2012). [CrossRef]

]. Unusual features relevant to the directional selectivity, which is inspired by the common effect of diffraction and dispersion, as in photonic crystal gratings, may include but are not restricted to one-way Rayleigh-Wood anomalies [11

11. A. E. Serebryannikov and E. Ozbay, “One-way Rayleigh-Wood anomalies and tunable narrowband transmission in photonic crystal gratings with broken structural symmetry,” Phys. Rev. A 87(5), 053804 (2013). [CrossRef]

], unidirectional splitting [20

20. E. Colak, A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Experimental study of broadband unidirectional splitting in photonic crystal gratings with broken structural symmetry,” Appl. Phys. Lett. 102(15), 151105 (2013). [CrossRef]

], and reflection-free diodelike transmission [3

3. A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]

]. On the other hand, it has been demonstrated in case of photonic crystal gratings that unidirectional transmission can be a purely diffraction effect, i.e., it can be achieved without the use of a peculiar dispersion [12

12. A. E. Serebryannikov, K. B. Alici, T. Magath, A. O. Cakmak, and E. Ozbay, “Asymmetric Fabry-Perot-type transmission in photonic-crystal gratings with one-sided corrugations at a two-way coupling,” Phys. Rev. A 86(5), 053835 (2012). [CrossRef]

,21

21. A. E. Serebryannikov, E. Colak, A. O. Cakmak, and E. Ozbay, “Dispersion irrelevant wideband asymmetric transmission in dielectric photonic crystal gratings,” Opt. Lett. 37(23), 4844–4846 (2012). [CrossRef] [PubMed]

]. Most of the known designs require structuring and/or assembling to obtain a proper configuration. Performances that can be easily fabricated and integrated with other components invoke materials whose properties could match the requirements to the asymmetric wave transmission without structuring. To this end, one should mention ultralow-ε (ε ranging from 0 to 1) and ultralow-index materials (refractive index ranging from 0 to 1) [22

22. B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B 20(12), 2448–2453 (2003). [CrossRef]

24

24. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]

]. For such a material, equifrequency dispersion contours have a circular shape and are narrower than in air. Thus, they enable unidirectional transmission. Nonsymmetric gratings based on Drude materials have been proposed [2

2. A. E. Serebryannikov and E. Ozbay, “Unidirectional transmission in non-symmetric gratings containing metallic layers,” Opt. Express 17(16), 13335–13345 (2009). [CrossRef] [PubMed]

,16

16. 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(11), 115111 (2006). [CrossRef]

]. However, the range of 0<ε<1 corresponds for most of metals to petahertz frequencies. At lower frequencies, Drude dispersion can be achieved by utilizing a wire medium, i.e., the structuring is still required [22

22. B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B 20(12), 2448–2453 (2003). [CrossRef]

].

2. Theoretical background

First, let us consider the general dispersion features of polar dielectrics in the connection with the conditions required for obtaining of unidirectional transmission.

Owing to phonon-photon coupling and polariton excitation, polar dielectrics behave as a strongly dispersive medium whose permittivity depends on frequency as follows [26

26. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter 49(16), 11080–11087 (1994). [CrossRef] [PubMed]

]:
εP(ω)=ε+(ε0ε)ωT2/(ωT2ω2iγω),
(1)
where ε0 is the static dielectric constant, ε is the high-frequency limit of dielectric constant, and γ is the absorption relevant factor. One of the basic features of polar dielectrics is the existence of the polaritonic gap, which corresponds to the range of εP<0, see Fig. 1(a).
Fig. 1 (a) Typical frequency dependence of permittivity of a polar dielectric εP at γ=0 and schematics of (b) two-layer and (c) single-layer nonsymmetric gratings containing polar dielectrics; for two- and single-layer gratings, D is the maximal thickness and L is grating period; for two-layer grating, h is thickness of the non-dispersive dielectric layer.
The frequencies of the lower and upper boundaries of this range are commonly denoted by ωT and ωL, respectively. They are connected by the Lyddane-Sachs-Teller relation [25

25. C. Kittel, Introduction to Solid State Physics (John Wiley, 2005).

]

ωL2/ωT2=ε0/ε.
(2)

To obtain dispersion relevant unidirectional transmission, the working frequency range should correspond to 0<εP<1, provided that the surrounding medium is air. One can see in Fig. 1(a) that the corresponding frequency range is located on the right of the polaritonic gap, whereas ω=ωL corresponds to εP=0 for γ=0. Clearly, the wider the frequency band, in which 0<εP<1, the wider the range of unidirectional transmission should be. The angular frequency value, at which εP(ω) achieves a desired value, ω=ωu, can easily be obtained from Eq. (1). For εP(ω)=1 and γ=0, it yields ωu2=(εωL2ωT2)/(ε1). Moreover, if the range of 0<εP<1/A, A>1, is considered as the preferable operation range, then
ωu2=(AεωL2ωT2)/(Aε1).
(3)
Accordingly, the expected width of the range of unidirectional transmission is estimated for given A as

Δω=ωuωL.
(4)

To compare different polar dielectrics from the point of view of their possible usage for achieving unidirectionality, we introduce the following dimensionless criterium:
ζ=Δω/ωL=[(AεωT2/ωL2)/(Aε1)]1/21.
(5)
It gives the relative width of the frequency band, in which unidirectional transmission is expected to appear. From Eq. (5), it is seen that the stronger the difference between ωT and ωL, and the smaller value of ε, the larger ζ is. The values of ωT, ωL, ε0 and ε can be found in the textbooks [25

25. C. Kittel, Introduction to Solid State Physics (John Wiley, 2005).

,37

37. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

]. However, it is worth noting that these values may differ from one book to another. The values of ωL/ωT, ε and ζ have been compared for various polar dielectrics. Results of the comparison are presented in Table 1, based on the data from Table 3 in Ref [25

25. C. Kittel, Introduction to Solid State Physics (John Wiley, 2005).

].

Table 1. Comparison of Various Polar Dielectrics in Terms of Characteristics Responsible for the Relative Width of Unidirectional Transmission Range

table-icon
View This Table
For the sake of definiteness, we take here A=2, i.e., 0<εP<0.5. One can see that there are materials, for which ωL/ωT>2 or/and ε<2, and thus the values of ζ can be rather large.

Clearly, the choice of a certain material depends not only on the values of ωL/ωT, ε and ζ but also on the frequency range targeted. Indeed, the values of ωT and ωL substantially differ from one polar dielectric to another, covering a wide frequency range, from units to several tens of terahertz. To better illustrate the possibility of wideband unidirectionality, we selected LiF. According to [27

27. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011). [CrossRef]

,37

37. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

], we use ωT=57.98THz, ωL=172.22THz, γ=4.4THz, and ε=1.04 for this material. Note that these values differ from those given for LiF and promise even a better performance as compared to that based on the data from [25

25. C. Kittel, Introduction to Solid State Physics (John Wiley, 2005).

].

To characterize asymmetry in transmission and reflection, we use T=(m)tm, T=(m)tm, R=(m)rm, and R=(m)rm, where T, T, R, and R mean corrugated-side (forward) transmittance, noncorrugated-side (backward) transmittance, corrugated-side reflectance, and noncorrugated-side reflectance, respectively, and tm, tm, rm, and rm mean the corresponding mth-order (partial) transmittances and reflectances. It is noteworthy that t0=t0=t0 because of the Lorentz reciprocity.

According to the general theory of the diffraction inspired unidirectional transmission [38

38. P. Rodríguez-Ulibarri, M. Beruete, M. Navarro-Cia, and A. E. Serebryannikov, “Wideband unidirectional transmission with tunable sign-switchable refraction and deflection in nonsymmetric structures,” Phys. Rev. B 88(16), 165137 (2013). [CrossRef]

], variation in εP can strongly affect the achievable regimes. In particular, the width of the θ range, in which unidirectional transmission is possible, depends on εP. As follows from the analysis of coupling for circular equifrequency dispersion contours in [38

38. P. Rodríguez-Ulibarri, M. Beruete, M. Navarro-Cia, and A. E. Serebryannikov, “Wideband unidirectional transmission with tunable sign-switchable refraction and deflection in nonsymmetric structures,” Phys. Rev. B 88(16), 165137 (2013). [CrossRef]

], the condition
εP1/2+1>2π/(kL),
(6)
where k=ω/c, is necessary for obtaining of unidirectional transmission at least within a narrow θ range, at given kL. In turn, this leads to restrictions on the value of A in Eq. (3). If εP1/2=π/(kL)=1/3, the range of θ, in which unidirectional transmission is expected to appear, extends from θ1=arcsin(1/3) to θ2=π/2 [38

38. P. Rodríguez-Ulibarri, M. Beruete, M. Navarro-Cia, and A. E. Serebryannikov, “Wideband unidirectional transmission with tunable sign-switchable refraction and deflection in nonsymmetric structures,” Phys. Rev. B 88(16), 165137 (2013). [CrossRef]

].

3. Basic features of transmission and reflection

In Fig. 2, forward transmittance and forward-to-backward transmittance contrast, CT=20log10T/T, are presented for the two configurations, which are expected to enable unidirectional transmission.
Fig. 2 T and tm vs kL for configurations (a) A and (b) B, and (c) CT vs kL, at ωTL/c=1.5π and θ=60°; (a) and (b): tm at m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, m=4 - black dashed line, and T - cyan dotted line; (c): CT for configurations A and B is shown by dark-green solid line and violet dashed line, respectively; gray rectangles - locations of the unidirectional transmission ranges.
A rather large value of θ has firstly been taken to ensure wideband unidirectionality. In the presented example, T>0, T0 and CT>70dB at least at 14.4<kL<19. This kL-range corresponds to 28.2THz<f<37.2THz, whereas L=24μm. In turn, the polaritonic gap extends from kL1.5π to kL14. Owing to the non-dispersive dielectric layer, the two-layer (two-fraction) grating is characterized by higher transmission than the one-layer (one-fraction) grating, but CT is higher for the latter. This is not a surprising feature, since the ultralow-ε material occupies a larger volume in this case. In Figs. 2(a) and 2(b), one can see that several diffraction orders contribute to T. In line with the grating theory [39

39. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).

], the mth order may propagate in air starting from
kL=kmL=2π|m|/(1sgnmsinθ).
(7)
In Fig. 2(a), the order m=2 appears first, so that Tt2 in the vicinity of kL=15. Then, the orders m=3, m=1, and m=4 appear at kL15.2, kL16.2, and kL17.1, respectively. Thus, the conventional order in appearance of higher orders that corresponds to Eq. (6) is broken here. In Fig. 2(b), Tt2 up to kL16, where the orders m=1 and m=3 start contributing to T. Since transmission is entirely suppressed within the considered kL-range at the noncorrugated-side illumination [see Fig. 2(c)], asymmetry in transmission occurs also in the actual mth-order frequency thresholds (wavelength cutoffs), i.e., they differ for the two utilized illumination directions.

On the contrary to Fig. 2, there is no anomaly in threshold frequencies (cutoff wavelengths), starting from (up to) which rm contributes to R. At the same time, Rr0>0.9 in the considered kL-range, for the both configurations A and B. Hence, reflections are not affected by the corrugations of the exit interface, i.e., the grating behaves in the reflection regime similarly to a noncorrugated ultralow-ε layer. As a result, strong asymmetry takes place in reflection and, in particular, in the frequency thresholds.

In Fig. 3, the dependencies of tm, T, and CT on kL are presented for the same parameter settings as in Fig. 2, but now for smaller values of θ.
Fig. 3 T and tm vs kL for configurations (a,d) A and (b,e) B, and (c,f) CT vs kL, at ωTL/c=1.5π, (a-c) θ=47° and (d-f) θ=40°; (a), (b), (d), (e): tm at m=0 - gray solid line, m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, m=4 - black dashed line, and T - cyan dotted line; (c,f): CT for configurations A and B is shown by dark-green solid line and violet dashed line, respectively; gray rectangles - unidirectional transmission ranges.
The effect of decrease of θ is well seen from the comparison of Figs. 2 and 3. In particular, some narrowing of the unidirectional transmission range, weaker difference between configurations A and B in terms of maxT, and increase of T for configuration B should be noticed. For the values of θ used in Fig. 3, unidirectionality is still a wideband effect. For a larger part of the unidirectional transmission range, higher contrasts are observed for configuration B. For instance, this takes place at kL=16.2 in Fig. 3(c), where Tt1+t20.2 and CT=120dB. Hence, directional selectivity can occur in a wide range of θ variation that enables tunability. Further decrease of θ results in that the order m=0 is coupled at smaller values of kL than in Fig. 3, in accordance with the general theory of the diffraction inspired unidirectional transmission [38

38. P. Rodríguez-Ulibarri, M. Beruete, M. Navarro-Cia, and A. E. Serebryannikov, “Wideband unidirectional transmission with tunable sign-switchable refraction and deflection in nonsymmetric structures,” Phys. Rev. B 88(16), 165137 (2013). [CrossRef]

]. This leads to that the unidirectionality disappears, provided that the coupling for zero order is not vanishingly weak.

4. Decreasing grating period and thickness

Now let us consider the basic effects arising due to decrease of ωTL/c at fixed other parameters. In fact, this means that we take a smaller value of L, and − since D/L is assumed to be fixed − also a smaller value of D. Shift of the unidirectional transmission range towards smaller values of kL, significant single-beam transmission due to lower nonzero diffraction orders, and change of the sequence in appearance of higher orders are among the expected effects. First, we take ωTL/c=1.1π that corresponds to the polaritonic gap ranging from kL=1.1π to kL=10.4 and L=17.9μm. In Fig. 4, an example of the dependencies of tm and T on kL is presented at the same parameters as in Fig. 2, except of the value of ωTL/c.
Fig. 4 T and tm vs kL for configurations (a) A and (b) B at ωTL/c=1.1π and θ=60°; tm at m=0 - gray solid line, m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, m=4 - black dashed line, and T - cyan dotted line; gray rectangles - unidirectional transmission ranges.
Comparing to Fig. 2, stronger forward transmission can be obtained for configuration A in the unidirectional regime. Thus, at large values of θ, a proper choice of ωTL/c can help to enhance performance for the two-layer (two-fraction) grating. At the same time, forward transmission for configuration B has not been substantially enhanced.

To compare, Fig. 5 presents the results for a smaller value of θ.
Fig. 5 T and tm for configurations (a) A and (b) B, and (c) T and tm for configuration B at ωTL/c=1.1π and θ=40°; tm and tm at m=0 - gray solid line, m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, m=4 - black dashed line, T and T - cyan dotted line; gray rectangles - unidirectional transmission ranges.
In turn, Figs. 5(a) and 5(b) differ from Figs. 3(d) and 3(e) only in the value of ωTL/c. The basic features relevant to the θ variation are similar to those observed at ωTL/c=1.5π. However, the narrowing of the unidirectional range, which is inspired by decrease of θ, is now much stronger pronounced. Comparing Figs. 5(b) and 5(c), one can see that unidirectional transmission with Tt1, maxt10.31 and T0 is obtained in the vicinity of kL=12.5. Hence, wideband unidirectionality with a rather high T can also be obtained for configuration B, i.e., when using a single-layer grating, at least at intermediate θ.

Next, we further decrease ωTL/c. In Fig. 6, tm, T and CT vs kL are presented at ωTL/c=0.7π and θ=60°.
Fig. 6 Same as Fig. 2 but for ωTL/c=0.7π; (a,b) gray solid line - t0.
Now, the polaritonic gap corresponds to 0.7π<kL<6.61 and L=11.4μm. For configuration A, T in the unidirectional regime is further enhanced. For configuration B, it remains low. Hence, at large θ, such a decrease of ωTL/c has sense only for configuration A. It is noteworthy that larger values of T in Fig. 6(a) are achieved at the price of decrease of CT, as follows from the comparison of Figs. 6(c) and 2(c). Moreover, CT decreases for configuration B, too. In the contrast with Fig. 2(c), CT is now higher for configuration A, over a larger part of the unidirectional transmission range. Asymmetry in reflection is also observed in this case. It manifests itself in the similar way as described above, e.g., Rr0 within a wide frequency range. For the sake of completeness, we consider diffraction at smaller values of θ and the same ωTL/c as in Fig. 6. Results are presented in Figs. 7 and 8.
Fig. 7 T and tm for configuration (a) A at θ=40°, (b) configuration B at θ=40°, and (c) configuration A at θ=47°, when ωTL/c=0.7π; tm at m=0 - gray solid line, m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, and T - cyan dotted line; gray rectangles - unidirectional transmission ranges.
Fig. 8 R and rm (a) and R and rm (b) vs kL for configuration B at ωTL/c=0.7π and θ=40°; rm and rm at m=0 - gray solid line, m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, R and R - cyan dotted line; (b): t0 and T are shown by thin gray solid line and thin cyan dotted line, respectively.
Single-beam unidirectional transmission with Tt1 is observed at 6.2<kL<7.1 and 7.35<kL<8.25 in Fig. 7(a), at 6.5<kL<7.5 in Fig. 7(b), and at 6.5<kL<8.5 in Fig. 7(c). To further increase maxT, additional optimization is required.

In Fig. 8, rm, rm, R and R vs kL are shown for the same case as in Fig. 7(b) and for the corresponding case of noncorrugated-side illumination. Besides, t0 and T are presented in Fig. 8(b). One can see that the contribution of rm to R strongly differs from that of rm to R. In particular, Rr0 over a wide kL-range, while all of the orders m=0,1,2,3 contribute to R. At kL>12, there are no significant reflections. At the same time, most part of the incident-wave energy can be converted into a reflected higher-order beam, as occurs in Fig. 8(a) at kL=5. A quite strong asymmetry in transmission can be possible even at smaller values of L and ωTL/c than in Figs. 2-8. As an example, Fig. 9 presents the results for ωTL/c=0.4π.
Fig. 9 T and tm (a) and T and tm (b) for configuration A at ωTL/c=0.4π and θ=60°, and same other parameters and notations as in Fig. 5.
Unidirectional transmission is observed in this case with maxTmaxt1>0.6 and T0 near kL=5.4. It is worth noting that the polaritonic gap is located here at 0.4π<kL<3.78 where L6.5μm.

For the comparison purposes, Fig. 10 presents the results for the grating, which differs from configuration A in that the ultralow-ε layer is made of NaCl.
Fig. 10 T and tm (a) and T and tm (b) vs kL for a similar configuration as A but with the ultralow-ε layer made of NaCl, at ωTL/c=2π and θ=60°; tm and tm at m=0 - gray solid line, m=1 - red dashed line, m=2 - green dotted line, m=3 - blue dash-dotted line, m=4 - black dashed line; T and T - cyan dotted line; gray rectangle - unidirectional transmission range.
For this material, ωT=30.9THz, ωL=50.37THz, γ=1.2THz, and ε=2.22 [27

27. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011). [CrossRef]

,37

37. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

]. In this case, the polaritonic gap extends from kL=2π up to kL10.2 and L=61μm. The range of unidirectional transmission is located at 10.4<kL<12 that approximately corresponds to 8.15THz<f<9.4THz. As expected, this range is narrower than for LiF, because of smaller ωL/ωT and larger ε. In spite of this, NaCl can be a reasonable choice when a lower frequency range is targeted.

5. Dispersion irrelevant asymmetry in transmission

Up to now, we considered asymmetric wave transmission connected with the common effect of dispersion and diffraction at inclined incidence. As known from the earlier works, asymmetry due to the common effect can be quite strong also at normal incidence [3

3. A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]

,20

20. E. Colak, A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Experimental study of broadband unidirectional splitting in photonic crystal gratings with broken structural symmetry,” Appl. Phys. Lett. 102(15), 151105 (2013). [CrossRef]

]. In particular, it might occur in the gratings based on photonic crystals or materials with hyperbolic type dispersion. However, this regime cannot be obtained in the non-structured configurations based on polar dielectrics, for which asymmetric wave transmission is associated with ultralow ε and, thus, with non-hyperbolic type of dispersion. On the other hand, recent studies show that asymmetry in transmission can be strongly pronounced, even if dispersion is not hyperbolic, i.e., it does not force zero order to be uncoupled [12

12. A. E. Serebryannikov, K. B. Alici, T. Magath, A. O. Cakmak, and E. Ozbay, “Asymmetric Fabry-Perot-type transmission in photonic-crystal gratings with one-sided corrugations at a two-way coupling,” Phys. Rev. A 86(5), 053835 (2012). [CrossRef]

,21

21. A. E. Serebryannikov, E. Colak, A. O. Cakmak, and E. Ozbay, “Dispersion irrelevant wideband asymmetric transmission in dielectric photonic crystal gratings,” Opt. Lett. 37(23), 4844–4846 (2012). [CrossRef] [PubMed]

]. Thus, it can be a fully diffraction effect. Furthermore, in this case, stronger transmission often appears at the noncorrugated-side illumination, on the contrary to the above-considered examples. Similar regimes have already been studied in the nonsymmetric dielectric photonic crystal gratings at θ=0 [12

12. A. E. Serebryannikov, K. B. Alici, T. Magath, A. O. Cakmak, and E. Ozbay, “Asymmetric Fabry-Perot-type transmission in photonic-crystal gratings with one-sided corrugations at a two-way coupling,” Phys. Rev. A 86(5), 053835 (2012). [CrossRef]

] and θ>0 [21

21. A. E. Serebryannikov, E. Colak, A. O. Cakmak, and E. Ozbay, “Dispersion irrelevant wideband asymmetric transmission in dielectric photonic crystal gratings,” Opt. Lett. 37(23), 4844–4846 (2012). [CrossRef] [PubMed]

]. Dependencies of tm, tm, T, T, and CT on kL are presented in Fig. 11, for the same configurations as in Figs. 2 and 3 but for θ=0.
Fig. 11 T and tm vs kL for configuration A (a), T and tm for configuration A (b), and CT for configurations A and B (c), at ωTL/c=1.5π and θ=0; (a) and (b): tm and tm at m=0 - gray solid line, m=±1 - red dashed line, m=±2 - green dotted line, m=±3 - blue dash-dotted line, and T and T - cyan dotted line; (c): CT for configurations A and B is shown by dark-green solid line and violet dashed line, respectively; asterisks indicate the cases of CT<20dB.
One can see that a well pronounced contrast can be achieved within several narrow frequency ranges for configuration A, whereas asymmetry in transmission is weakly pronounced for the corresponding configuration B. For example, this situation occurs at kL=16.5, i.e., for one of the dips of CT<20dB in Fig. 11(c), where T=0.4 and t1=0.18. It is expected that this regime of asymmetric wave transmission can be obtained for various polar dielectrics, including those with a narrow range of ultralow ε, see Table 1. Indeed, if it is not necessary that zero order is formally uncoupled, the requirement of a wide range of 0<εP<1/A, A>1 can be cancelled. Thus, more polar dielectrics can be considered as viable candidates to design gratings that are similar to the configurations A and B. In Fig. 12, an example of the dependencies of tm, tm, T, T, and CT on kL is presented for the grating that differs from configuration B only in the used polar dielectric.
Fig. 12 T and tm (a), T and tm (b), and CT (c) vs kL, for configuration that is similar to B but made of GaAs, at ωTL/c=2π and θ=0; (a) and (b): tm and tm at m=0 - gray solid line, m=±1 - red dashed line, and T and T - cyan dotted line; (c): asterisk indicates the case of CT<20dB.
Here, it is GaAs, which has ωT=8.12THz, ωL=8.75THz, γ=0.5THz, and ε=10.9 [26

26. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter 49(16), 11080–11087 (1994). [CrossRef] [PubMed]

].

At the chosen value of ωTL/c, the polaritonic gap is located at 2π<kL<6.77. Since ωL/ωT1.08 and ε is rather large, wideband unidirectional transmission due to the common effect of dispersion and diffraction is not expected to appear. However, as seen in Fig. 12, asymmetry in transmission can be quite strong within a narrow range near kL=9.2, where Tt1+t+1>0.3. Hence, neither large ωL/ωT nor small ε is necessary for obtaining of a strongly pronounced asymmetry.

6. Conclusions

Merging the recently developed asymmetric wave transmission theory with the peculiar dispersion of polar dielectrics creates a proper platform to realize wideband directional selectivity at terahertz frequencies. In spite of that nonreciprocal operation regimes cannot be achieved in this framework because additional transmission channels (higher orders in our case) are needed to accept the re-directed energy, it looks very promising for manipulation of terahertz waves. Based on the obtained results, the main features of asymmetry arising in transmission and reflection have been demonstrated for the nonsymmetric gratings, which contain polar dielectrics working in the ultralow-ε regime inspired by phonon-photon coupling. The best pronounced asymmetry results from the common effect of asymmetric coupling and diffractions at the two interfaces, on the one hand, and the ultralow-ε relevant dispersion features, on the other hand. It can be obtained at large and intermediate angles of incidence. Adding a high-ε corrugated layer is required for obtaining of significant transmission at large angles, while both single-fraction and two-fraction nonsymmetric grating can be used at intermediate angles. However, transmission can be quite strong for one of the two incidence directions and can vanish for the opposite direction, even if dispersion does not force zero order, which is responsible for the symmetric transmission component, to be uncoupled. This regime is especially important for small angles, at which the common effect based mechanism is not possible. It can be achieved both with and without the use of the high-ε corrugated layer, depending on the chosen material and geometrical parameters. Hence, co-existence of dispersion relevant and irrelevant unidirectional regimes is not a unique feature of more complex nonsymmetric structures, e.g., photonic crystal gratings. The main regimes of such structured configurations can be replicated in simpler configurations but optimization can be required in order to achieve high transmittance for one of the two directions. Although only two types of simple nonsymmetric gratings have been considered here, more advanced structures could be suggested that exploit the same mechanism. Adjustment of thickness-to-period ratio allow us controlling contribution of desired and unwanted diffraction orders within the unidirectional range.

Acknowledgments

This work is supported by the projects DPT-HAMIT, ESF-EPIGRAT, NATO-SET-181, and by TUBITAK under Project Nos., 107A004, 109A015, 109E301. Contribution of A.E.S. has partially been supported by the Matsumae International Foundation (MIF), Japan under Research Fellowship Program. E.O. acknowledges partial support from the Turkish Academy of Sciences.

References and links

1.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 74(5), 056611 (2006). [CrossRef] [PubMed]

2.

A. E. Serebryannikov and E. Ozbay, “Unidirectional transmission in non-symmetric gratings containing metallic layers,” Opt. Express 17(16), 13335–13345 (2009). [CrossRef] [PubMed]

3.

A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]

4.

X.-B. Kang, W. Tan, Z.-S. Wang, Z.-G. Wang, and H. Cheng, “High-efficiency one-way transmission by one-dimensional photonic crystal with gratings on one side,” Chin. Phys. Lett. 27(7), 074204 (2010). [CrossRef]

5.

M. Stolarek, D. Yavorskiy, R. Kotyński, C. J. Zapata Rodríguez, J. Łusakowski, and T. Szoplik, “Asymmetric transmission of terahertz radiation through a double grating,” Opt. Lett. 38(6), 839–841 (2013). [CrossRef] [PubMed]

6.

A. Cicek, M. B. Yucel, O. A. Kaya, and B. Ulug, “Refraction-based photonic crystal diode,” Opt. Lett. 37(14), 2937–2939 (2012). [CrossRef] [PubMed]

7.

S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “Spoof-plasmon relevant one-way collimation and multiplexing at beaming from a slit in metallic grating,” Opt. Express 20(24), 26636–26648 (2012). [CrossRef] [PubMed]

8.

V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012). [CrossRef] [PubMed]

9.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013). [CrossRef]

10.

S. Xu, C. Qiu, and Z. Liu, “Acoustic transmission through asymmetric grating structures made of cylinders,” J. Appl. Phys. 111(9), 094505 (2012). [CrossRef]

11.

A. E. Serebryannikov and E. Ozbay, “One-way Rayleigh-Wood anomalies and tunable narrowband transmission in photonic crystal gratings with broken structural symmetry,” Phys. Rev. A 87(5), 053804 (2013). [CrossRef]

12.

A. E. Serebryannikov, K. B. Alici, T. Magath, A. O. Cakmak, and E. Ozbay, “Asymmetric Fabry-Perot-type transmission in photonic-crystal gratings with one-sided corrugations at a two-way coupling,” Phys. Rev. A 86(5), 053835 (2012). [CrossRef]

13.

M. Beruete, A. E. Serebryannikov, V. Torres, M. Navarro-Cia, and M. Sorolla, “Toward compact millimeter-wave diode in thin stacked-hole array assisted by a dielectric grating,” Appl. Phys. Lett. 99(15), 154101 (2011). [CrossRef]

14.

M. Mutlu, S. Cakmakyapan, A. E. Serebryannikov, and E. Ozbay, “One-way reciprocal spoof surface plasmons and relevant reversible diodelike beaming,” Phys. Rev. B 87(20), 205123 (2013). [CrossRef]

15.

W.-M. Ye, X.-D. Yuan, C. C. Guo, and C. Zen, “Unidirectional transmission in non-symmetric gratings made of isotropic material,” Opt. Express 18(8), 7590–7595 (2010). [CrossRef] [PubMed]

16.

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(11), 115111 (2006). [CrossRef]

17.

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009). [CrossRef]

18.

C. Wang, X.-L. Zhong, and Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012). [PubMed]

19.

J. H. Oh, H. W. Kim, P. S. Ma, H. M. Seung, and Y. Y. Kim, “Inverted bi-prism phononic crystals for one-sided elastic wave transmission applications,” Appl. Phys. Lett. 100(21), 213503 (2012). [CrossRef]

20.

E. Colak, A. E. Serebryannikov, A. O. Cakmak, and E. Ozbay, “Experimental study of broadband unidirectional splitting in photonic crystal gratings with broken structural symmetry,” Appl. Phys. Lett. 102(15), 151105 (2013). [CrossRef]

21.

A. E. Serebryannikov, E. Colak, A. O. Cakmak, and E. Ozbay, “Dispersion irrelevant wideband asymmetric transmission in dielectric photonic crystal gratings,” Opt. Lett. 37(23), 4844–4846 (2012). [CrossRef] [PubMed]

22.

B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B 20(12), 2448–2453 (2003). [CrossRef]

23.

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 70(4), 046608 (2004). [CrossRef] [PubMed]

24.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]

25.

C. Kittel, Introduction to Solid State Physics (John Wiley, 2005).

26.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter 49(16), 11080–11087 (1994). [CrossRef] [PubMed]

27.

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011). [CrossRef]

28.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543–545 (2004). [CrossRef]

29.

P. B. Catrysse and S. Fan, “Near-complete transmission through subwavelength hole arrays in phonon-polaritonic thin films,” Phys. Rev. B 75(7), 075422 (2007). [CrossRef]

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A. Rung, C. G. Ribbing, and M. Qiu, “Gap maps for triangular photonic crystals with a dispersive and absorbing component,” Phys. Rev. B 72(20), 205120 (2005). [CrossRef]

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S. Nojima, “Excitonic polaritons in one-dimensional photonic crystals,” Phys. Rev. B 57(4), R2057–R2060 (1998). [CrossRef]

32.

S. Nojima, “Photonic-crystal laser mediated by polaritons,” Phys. Rev. B 61(15), 9940–9943 (2000). [CrossRef]

33.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70(4), 1137–1144 (2001). [CrossRef]

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M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Spectral properties of exciton polaritons in one-dimensional resonant photonic crystals,” Phys. Rev. B 73(11), 115321 (2006). [CrossRef]

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S. Nojima, “Optical response of excitonic polaritons in photonic crystals,” Phys. Rev. B 59(8), 5662–5677 (1999). [CrossRef]

36.

T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22(11), 2405–2418 (2005). [CrossRef] [PubMed]

37.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

38.

P. Rodríguez-Ulibarri, M. Beruete, M. Navarro-Cia, and A. E. Serebryannikov, “Wideband unidirectional transmission with tunable sign-switchable refraction and deflection in nonsymmetric structures,” Phys. Rev. B 88(16), 165137 (2013). [CrossRef]

39.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(050.1970) Diffraction and gratings : Diffractive optics
(120.7000) Instrumentation, measurement, and metrology : Transmission
(160.4670) Materials : Optical materials

ToC Category:
Terahertz Optics

History
Original Manuscript: December 2, 2013
Revised Manuscript: January 20, 2014
Manuscript Accepted: January 21, 2014
Published: February 3, 2014

Citation
Andriy E. Serebryannikov, Ekmel Ozbay, and Shunji Nojima, "Asymmetric transmission of terahertz waves using polar dielectrics," Opt. Express 22, 3075-3088 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3075


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References

  1. M. J. Lockyear, A. P. Hibbins, K. R. White, J. R. Sambles, “One-way diffraction grating,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 74(5), 056611 (2006). [CrossRef] [PubMed]
  2. A. E. Serebryannikov, E. Ozbay, “Unidirectional transmission in non-symmetric gratings containing metallic layers,” Opt. Express 17(16), 13335–13345 (2009). [CrossRef] [PubMed]
  3. A. E. Serebryannikov, A. O. Cakmak, E. Ozbay, “Multichannel optical diode with unidirectional diffraction relevant total transmission,” Opt. Express 20(14), 14980–14990 (2012). [CrossRef] [PubMed]
  4. X.-B. Kang, W. Tan, Z.-S. Wang, Z.-G. Wang, H. Cheng, “High-efficiency one-way transmission by one-dimensional photonic crystal with gratings on one side,” Chin. Phys. Lett. 27(7), 074204 (2010). [CrossRef]
  5. M. Stolarek, D. Yavorskiy, R. Kotyński, C. J. Zapata Rodríguez, J. Łusakowski, T. Szoplik, “Asymmetric transmission of terahertz radiation through a double grating,” Opt. Lett. 38(6), 839–841 (2013). [CrossRef] [PubMed]
  6. A. Cicek, M. B. Yucel, O. A. Kaya, B. Ulug, “Refraction-based photonic crystal diode,” Opt. Lett. 37(14), 2937–2939 (2012). [CrossRef] [PubMed]
  7. S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, E. Ozbay, “Spoof-plasmon relevant one-way collimation and multiplexing at beaming from a slit in metallic grating,” Opt. Express 20(24), 26636–26648 (2012). [CrossRef] [PubMed]
  8. V. Liu, D. A. B. Miller, S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012). [CrossRef] [PubMed]
  9. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013). [CrossRef]
  10. S. Xu, C. Qiu, Z. Liu, “Acoustic transmission through asymmetric grating structures made of cylinders,” J. Appl. Phys. 111(9), 094505 (2012). [CrossRef]
  11. A. E. Serebryannikov, E. Ozbay, “One-way Rayleigh-Wood anomalies and tunable narrowband transmission in photonic crystal gratings with broken structural symmetry,” Phys. Rev. A 87(5), 053804 (2013). [CrossRef]
  12. A. E. Serebryannikov, K. B. Alici, T. Magath, A. O. Cakmak, E. Ozbay, “Asymmetric Fabry-Perot-type transmission in photonic-crystal gratings with one-sided corrugations at a two-way coupling,” Phys. Rev. A 86(5), 053835 (2012). [CrossRef]
  13. M. Beruete, A. E. Serebryannikov, V. Torres, M. Navarro-Cia, M. Sorolla, “Toward compact millimeter-wave diode in thin stacked-hole array assisted by a dielectric grating,” Appl. Phys. Lett. 99(15), 154101 (2011). [CrossRef]
  14. M. Mutlu, S. Cakmakyapan, A. E. Serebryannikov, E. Ozbay, “One-way reciprocal spoof surface plasmons and relevant reversible diodelike beaming,” Phys. Rev. B 87(20), 205123 (2013). [CrossRef]
  15. W.-M. Ye, X.-D. Yuan, C. C. Guo, C. Zen, “Unidirectional transmission in non-symmetric gratings made of isotropic material,” Opt. Express 18(8), 7590–7595 (2010). [CrossRef] [PubMed]
  16. A. E. Serebryannikov, T. Magath, K. Schuenemann, O. Y. Vasylchenko, “Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials,” Phys. Rev. B 73(11), 115111 (2006). [CrossRef]
  17. R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009). [CrossRef]
  18. C. Wang, X.-L. Zhong, Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012). [PubMed]
  19. J. H. Oh, H. W. Kim, P. S. Ma, H. M. Seung, Y. Y. Kim, “Inverted bi-prism phononic crystals for one-sided elastic wave transmission applications,” Appl. Phys. Lett. 100(21), 213503 (2012). [CrossRef]
  20. E. Colak, A. E. Serebryannikov, A. O. Cakmak, E. Ozbay, “Experimental study of broadband unidirectional splitting in photonic crystal gratings with broken structural symmetry,” Appl. Phys. Lett. 102(15), 151105 (2013). [CrossRef]
  21. A. E. Serebryannikov, E. Colak, A. O. Cakmak, E. Ozbay, “Dispersion irrelevant wideband asymmetric transmission in dielectric photonic crystal gratings,” Opt. Lett. 37(23), 4844–4846 (2012). [CrossRef] [PubMed]
  22. B. T. Schwartz, R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B 20(12), 2448–2453 (2003). [CrossRef]
  23. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 70(4), 046608 (2004). [CrossRef] [PubMed]
  24. A. Alù, M. G. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
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