Unidirectional transmission realized by two nonparallelgratings made of isotropic media

Wei-Min Ye,* Xiao-Dong Yuan, and Chun ZengCollege of Optoelectics Science and Engineering, National University of Defense Technology, Changsha, 410073, China

*Corresponding author: wmye72@126.com

Received May 16, 2011; revised June 17, 2011; accepted June 26, 2011;posted June 27, 2011 (Doc. ID 147567); published July 22, 2011

We realize a unidirectional transmission by cascading two nonparallel gratings (NPGs) made of isotropic, lossless,and linear media. For a pair of orthogonal linear polarizations, one of the gratings is designed as a polarizer, which isa reflector for one polarization and a transmitter for the other; another grating is designed as a polarization converter,which converts most of one polarized incident wave into another polarized transmitted wave. It is demonstrated bynumerical calculation that more than 85% of the incident light energy can be transmitted with less than 1% trans-mission in the opposite direction for linearly polarized light at normal incidence, and the relative bandwidth of theunidirectional transmission is nearly 9%. The maximum transmission contrast ratio between the two directions is62dB. Unlike one-way diffraction grating, the transmitted light of the NPGs is collinear with the incident light, buttheir polarizations are orthogonal. © 2011 Optical Society of AmericaOCIS codes: 050.1970, 050.1950, 120.7000.

The unidirectional light transmission means that certainmodes are allowed to propagate in only one direction,while propagation in the opposite direction is prohibitedor significantly suppressed. Based on the reciprocityprinciples in optics [1], we know that there exist threegenerally asymmetric systems that are possible to realizea unidirectional light transmission: (1) a magnetoopticmedium, which breaks the time reversal symmetry ofthe asymmetric system [2]; (2) a nonlinear medium,which ensures unidirectional transmission only for thenonlinear transmission [3]; and (3) a mode converter inthe asymmetric structure composed of the linear and iso-tropic media. Unlike the first and second systems, theunidirectional light propagation in this case is accompa-nied by an asymmetric mode conversion and does notviolate Lorentz’s reciprocity theorem [4–6].In free space, different polarizations and propagation

directions belong to different modes. Thus, mode con-version in free space can be realized by polarizationconversion and propagation direction inversion. Theasymmetric transmission is achieved in metamaterialstructures that support asymmetric polarization conver-sion. For example, the normal incidence transmission ofcircularly polarized light through a lossy anisotropicplanar chiral metamaterial could be asymmetric in theopposite directions [4,5]. A genuine three-dimensionalchiral metamaterial [6] with structural variation in theprincipal propagation direction could exhibit asymmetrictransmission for linearly polarized light. On the otherhand, based on asymmetric conversion between thezeroth- and nonzeroth-order diffraction of lights, one-way diffraction gating (OWDG) [7] with different periodsof the front–side and back–side interfaces could realizeunidirectional light propagation. To increase the contrastratio between forward and backward transmittances,photonic crystals [8] with nonzero directional bandgapcombined with subwavelength grating [9] as a widebandreflector are introduced in OWDG to eliminate the reci-procal zeroth-order propagation. Therefore, the propaga-tion direction of unidirectionally transmitted light of theOWDG is different with the incident light.

In this Letter, we theoretically demonstrate a novelunidirectional transmission for linearly polarized lightrealized by asymmetric polarization conversion. The de-vice is made of isotropic, lossless, and linear media. Thesystem is made up of two cascaded subwavelength non-parallel gratings (NPGs). For a pair of orthogonal linearpolarizations, one of the gratings is designed as a polar-ization converter, which could be chosen to be sym-metric for both polarizations. The other is designed asa polarizer, which is transparent only for one of the po-larizations and ensures the asymmetric propagation oflinear polarizations.

As an example, we propose NPGs made with silicon(Si) gratings buried in SiO2 because of their fabricationcompatibility with the standard complementary metal–oxide–semiconductor technology and low absorptioncoefficient in the near-IR spectral region. A general struc-ture of the NPGs and the definition of the coordinate sys-tem are shown in Figs. 1(a) and 1(b). The subwavelengthSi=SiO2 grating (period a1, thickness h1, filling factor of Sif 1 ¼ w1=a1) on the bottom, assumed to be infinite in thex direction and periodic in the y direction, is designedas a polarizer for x or y polarization. The other Si=SiO2grating (period a2, thickness h2, filling factor of Sif 2 ¼ w2=a2) above, assumed to be direction rotated byan angle of θ to the first grating, works as a polarizationconverter for x and y polarization. They are separated inthe z direction by a homogeneous SiO2 layer (thicknessd2) on the SiO2 substrate (thickness d1) and covered byanother homogeneous SiO2 layer (thickness d3). Here,we focus only on the transmission of normal incident

Fig. 1. Schematic of the proposed NPGs and the definition ofthe coordinate system.

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and linearly polarized light through three different struc-tures of gratings and limit the frequency of the incidentlight to the range that only the zeroth-order diffraction ofthe gratings can propagate. Because the designed NPGsis two-dimensional periodical in the x–y plane, the scat-tering matrix method based on plane wave expansion isused to study the transmission spectrum. In our calcula-tion, the relative dielectric constant of Si, SiO2, and airare chosen to be 11.56, 2.13, and 1.0, respectively.The first structure we studied is a single-layer Si=SiO2

grating (period a, thickness h, filling factor of Si f , infinitein the x direction and periodic in the y direction) with thesemi-infinite SiO2 as the substrate and the cover. Basedon the effects of leaky modes of the single-layer gratingtransmission [10], it is not difficult to design a single-layerthin Si=SiO2 grating as a wideband reflector for the xpolarization (or TE mode) [9,10]. To achieve a high trans-mission of the y polarization (or TM mode), a relativelysimple method is to decease the filling factor of Si in theSi=SiO2 grating. Figure 2 shows the normal incidenttransmittance spectra of linearly polarized light througha single-layer Si=SiO2 grating (period a, thickness h ¼0:37a, filling factor of Si f ¼ 0:2). There exists a high-reflectance band of the x polarization with thenormalized frequency (ωa=ð2πcÞ) of incoming light fall-ing in the range of ½0:577; 0:634�, where the transmittanceof x polarization is less than 0.01, but that of y polariza-tion is larger than 0.99. Thus, this Si=SiO2 grating couldwork efficiently as a polarizer.To design a polarization converter, both the transmit-

tances and phase difference [11] of x and y polarizationthrough a single-layer grating have to be considered. Inanalogy to a slab of homogeneous and anisotropic med-ium with two principal axes parallel to x and y, respec-tively, the polarization of the transmitted light will be thesuperposition of its x and y polarizations. We assume alinear polarized incident light with its polarization direc-tion at an angle of α to the x direction. Its transmissioncoefficients are denoted by real number tx and complexnumber tyeiω for x and y polarizations, respectively,where φ is the phase difference of the two orthogonallinear polarizations. If we define the energy efficiencyTeff as the transmitted energy ratio converted to the po-larization that is orthogonal to the incident polarization,then Teff can be expressed in terms of tx, ty, and φ as

Teff ¼14sin2ð2αÞðt2x þ t2y − 2txty cosφÞ: ð1Þ

From Eq. (1), we know that a single-layer grating, as apolarization converter for x and y-polarization, mustbe high transmittance for both polarizations and havea phase difference φ nearly equal to 180°. Therefore,we choose a single-layer thick grating. Figure 3(a) pre-sents the transmission spectra at normal incidence oflinearly polarized light through a single-layer Si=SiO2grating (period a, thickness h ¼ 0:8a, filling factor ofSi f ¼ 0:16). Because of the excitation of the leaky mode,there is one transmission dip for each of the linear polar-izations, the central normalized frequency of the dips are0.562 (x polarizations) and 0.664 (y polarizations), re-spectively. Between these two dips, the transmittancesof the pair of linear polarizations with the normalizedfrequency falling in the interval ½0:572; 0:661� are largerthan 0.9. In addition, Fig. 3(b) shows that, in this fre-quency interval, the phase difference φ between the pairof linear polarizations ranges from 151° to 199°, and theefficient transmittance Teff with the angle α equal to 45°is larger than 0.9. Based on these results, we can realize apolarization converter by a single-layer grating.

The second structure is a single-layer Si=SiO2 grating(period a, thickness h ¼ 0:8a, filling factor of Si f ¼ 0:16,infinite in the direction at 45° to the x direction and per-iodic in the direction at 135° to the x direction) with thesemi-infinite SiO2 as the substrate and the cover. Similarto an anisotropic medium layer, the transmitted lightof a normal incident x polarization through the gratingcomposes both x and y polarizations. Figure 4 showsthe relative polarization transmittance spectra of a nor-mal incident x polarization through the grating, wherethe transmittances T ðyxÞ and T ðxxÞ denote the y- and x-polarized transmittance of an incident x polarization,

Fig. 2. Normal incident transmittance spectra of the linearlypolarized light through a single-layer Si=SiO2 grating (perioda, thickness h ¼ 0:37a, filling factor of Si f ¼ 0:2, infinite inthe x direction and periodic in the y direction).

Fig. 3. (a) Normal incident transmittance spectra and(b) phase difference and efficient transmittance definedby Eq. (1) of x and y polarization through a single-layerSi=SiO2 grating (period a, thickness h ¼ 0:8a, filling factor ofSi f ¼ 0:16, infinite in the x direction and periodic in the ydirection).

Fig. 4. Relative polarization transmittance spectra of a normalincident x polarization through a single-layer Si=SiO2 grating(period a, thickness h ¼ 0:8a, filling factor of Si f ¼ 0:16, infi-nite in the direction at 45° to the x direction and periodic in thedirection at 135° to the x direction).

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respectively. It is obvious that T ðyxÞ of normalized fre-quency among the interval ½0:573; 0:661� is larger than0.9. On the other hand, T ðxxÞ of normalized frequencyamong the interval ½0:59; 0:625� is less than 0.01. There-fore, consistent with the result shown by Fig. 3(b), thissingle-layer Si=SiO2 grating is an efficient polarizationconverter of x and y polarizations.Combining these two structures, we get the last struc-

ture: the proposed NPGs shown in Figs. 1(a) and 1(b).Based on the results shown in Figs. 2 and 4, we chosethe parameters of two cascaded nonparallel Si=SiO2gratings as period a1 ¼ a2 ¼ a, thickness h1 ¼ 0:37a,h2 ¼ 0:8a, filling factors f 1 ¼ 0:2, f 2 ¼ 0:16 and the angleθ ¼ 45°. To reduce the influence of the coupling betweenthese two gratings and the finite thickness of SiO2 sub-strate and cover on the performance of NPGs, we chooserelative large thickness for these three homogeneousSiO2 layers, i.e., d1 ¼ d3 ¼ 16a, d2 ¼ 2a. Figure 5 showsthe relative polarization transmittance spectra of a nor-mal incident x and y polarization traveling in negativez direction (5(a) and 5(c)) and positive z direction(5(b) and 5(d)) through the proposed NPGs, whereT ðxyþÞ and T ðxy−Þ denote the x-polarized transmittanceof a normal incident y polarization traveling in positiveand negative z directions, respectively, and so asT ðyyþÞ, T ðxxþÞ, T ðyxþÞ, T ðyy−Þ, T ðyx−Þ, and T ðxx−Þ. It isobvious in Fig. 5 that these relative polarization transmit-tances are not independent, and they satisfy T ðyxþÞ ¼T ðxy−Þ, T ðxyþÞ ¼ T ðyx−Þ, T ðxxþÞ ¼ T ðxx−Þ, and T ðyyþÞ ¼T ðyy−Þ. The four equality relationships are in accordancewith the reciprocity principles in optics [4,6]. However,Figs. 5(a) and 5(b) show unidirectional transmission

(T ðyx−Þ larger than 0.85, T ðyxþÞ þ T ðxxþÞ less than 0.01)of a normal incident x polarization with normalized fre-quency among the interval ½0:58; 0:635� through the pro-posed NPGs. The maximum transmission contrastratio between transmittances of positive and negativez direction reaches 62 dB at normalized frequency0.615. The unidirectional transmission is caused by theasymmetric polarization conversion in the NPGs, whichis expressed quantitatively by the difference betweenT ðyx−Þ and T ðxy−Þ. In addition, due to the finite thicknessof the NPGs, significant Fabry–Perot oscillation is ob-served in the spectrum of T ðyx−Þ and T ðxyþÞ.

In conclusion, we have proved theoretically that a uni-directional transmission of linearly polarized light couldbe realized by NPGs made of isotropic, lossless, and lin-ear media. The key idea for the unidirectional transmis-sion in NPGs is an asymmetric polarization conversionachieved by cascading a polarizer with a polarizationconverter. High unidirectional transmittance and trans-mission contrast ratio between opposite directions ofpropagations are two inherent advantages of NPGs madeby lossless and linear media. Unlike OWDG, polarizationdirections of the incident and transmitted light of NPGsare orthogonal, but propagation directions are parallelbecause only the zeroth-diffraction order of gratingsis designed to propagate. Furthermore, the proposedNPGs are not sensitive to the relative position of thetwo gratings, which presents an obvious advantage forfabrication.

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Fig. 5. (a), (c) Relative polarization transmittance spectra of anormal incident x and y polarization traveling through the pro-posed NPGs in the negative z direction and (b), (d) that in thepositive z direction.

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