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We achieve a broadband unidirectional transmission or One-way diffraction grating by cascading two parallel gratings made of isotropic material with different periods. In order to significantly reduce the reciprocal transmission of the zero order, one of them is chosen to be a subwavelength grating and designed as a wideband reflector for the incident-wave. It is demonstrated that more than 65 percent of the incident-wave energy can be transmitted unidirectionally with less than 0.22 percent transmission in the opposite direction at normal incidence for TE polarization. And, the relative bandwidth of the unidirectional transmission is greater than 10 percent.
©2010 Optical Society of America
1. Introduction
Unidirectional light propagation is an interesting electromagnetic phenomenon, which is connected with breaking time-reversal symmetry or inversion symmetry. The key to break time-reversal symmetry is that at least one of the constitutive components is a non-reciprocal (Faraday-effect) and anisotropic material. It is proved that reflect-free one-way waveguide modes, whose group velocities are only along one direction, could exist at the interface of certain 2D photonic crystals consisting of gyromagnetic [1] and gyroelectric [2] media. On another hand, based on the anomalous diffraction effects of the non-symmetric isotropic material gratings, with different periods of the front-side and back-side interfaces, unidirectional light propagation could also be realized. This kind of non-symmetric grating is called as a “one-way diffraction gating (OWDG) ” [3]. The physical basis of OWDG is to eliminate the zero order propagation in non-symmetric gratings, which is responsible for reciprocal transmission. To do it, photonic crystals [4] with non-zero directional band gap and metal layer [5]with a small positive real part of the index of refraction are inserted in non-symmetric gratings. But the narrowband of photonic crystals layer and the loss of metal layer limit the potential application of OWDG.
Subwavelength gratings consisting of isotropic dielectric have attracted intensive research for many years, because subwavelength resonant leaky modes in their periodic layer can efficiently control the transmission spectra of the gratings [6]. For example, wideband reflectors [7,8] using a subwavelength grating have been reported in experiments. Comparing with Bragg reflectors operating on the basis of multilayer thin films, subwavelength gratings are compact and can complement or enhance the functionality and applicability of thin films [6].
In this paper, we utilize a subwavelength grating as a wideband reflector to reduce the zero order propagation in non-symmetric gratings, and obtain a wideband unidirectional transmission by cascading it with another isotropic material grating of different periods.
2. Design
We propose an OWDG made with periodic silicon (Si) layer and SiO2 layer on the SiO2 substrate, because of their fabrication compatibility with the standard CMOS technology and low absorption coefficient in the near infrared spectral region. A general structure of the OWDG is shown in Fig. 1 . That is, a subwavelength Si/ SiO2 grating (period a, thickness h, filling factor of Si f = w/a) buried beneath a SiO2 /air diffraction grating (period a 2, thickness h 2, filling factor of SiO2 f 2 = w 2/a 2) spaced by a homogeneous SiO2 layer(thickness d) in the between. The Si/SiO2 grating and SiO2/air gratings are designed in such a way that if the light incomes from the SiO2 substrate at certain range of incidence angle around normal incidence, the subwavelength Si/SiO2 grating will reflect it as a wideband reflector [6], if the light incomes from the air at a normal incidence, it will be diffracted into zero and non-zero orders by SiO2/air grating. The non-zero order transmission lights in homogeneous SiO2 layer can propagate into the substrate because its incidence angle is out of the range in which the Si/SiO2 grating operates as a reflector. So, unidirectional transmission of the OWDG requires that there exist k xm, the wave number along x axis of a non-zero mth-order diffraction light generated by SiO2 /air grating, which satisfies
Where, k x0 and ω are the wave number along x axis and frequency of an incident wave which is reflected totally by Si/ SiO2 grating, θc is maxim incidence angle of the Si/SiO2 grating as a reflector, is the refractive index of SiO2.For the sake of simplicity, it is assumed that the gratings are infinite in the y direction, and that the materials are lossless and dispersion free, and the incident light is in the xoz plane. So, the TE polarization (E x = E z = H y = 0) and TM polarization (H x = H z = E y = 0) are uncoupled. We limit the study only to TE polarization and period a 2 equal to 2a as shown in Fig. 1. That is, 2a is the period of the OWDG. Therefore, scattering matrix method based on plane-wave expansion [9] is used to study transmission spectral of the OWDG. We focus on the unidirectional transmission at normal incidence. The relative dielectric constant of Si, SiO2 and air in the calculation are chosen to be 11.56, 2.13 and 1.0, respectively. The thickness and filling factor of the two gratings layers are the major design parameters.
3. Numerical Simulation
At first, we study the performance of the subwavelength Si/SiO2 grating (the substrate and the cover layer are SiO2). The parameters that are optimized to achieve the high reflectance are thickness h = 0.3a, filling factor f = 0.36. Figure 2(a) and (b) show the reflectance and transmittance spectra of the Si/SiO2 grating at normal incidence for TE polarization respectively. When the normalized frequency (ωa/2πc) of incoming light is within the high-reflectance band [0.572, 0.653], the reflectance in Fig. 2(a) is larger than 0.99. This high-reflection band of Si/SiO2 grating is supported by a blend of two leaky modes [6], which correspond to the two transmittance dips whose normalized frequency are 0.589 and 0.637 respectively in Fig. 2(b).
Fig. 2 Reflectance (a) and transmittance (b) spectra of the subwavelength Si/SiO2 grating (thickness h = 0.3a, filling factor f = 0.36) at normal incidence for TE polarization.
To obtain the range of incidence angle, in which the Si/SiO2 grating operates as a reflector, we calculate the transmittance spectra of the Si/SiO2 grating with different incidence angle. Figure 3 shows the transmittance spectra of the TE polarization with the incident angle θ equals to 0°, 2°, 5° and 10° respectively. From this figure, we can see that the high-reflectance band (the reflectance larger than 0.99) becomes narrower with the increase of the incident angle, and disappears when the incident angle θ increase to 5°. The transmittance of Si/SiO2 grating is larger than 0.49 in the high-reflectance band [0.572, 0.653] when the incident angle θ is equal to 10°.
Fig. 3 Transmittance spectra of the subwavelength Si/SiO2 grating for the TE polarization with the incident angle θ = 0°, 2°, 5° and 10°.
Secondly, we consider the diffraction efficiency of the SiO2/air grating (the substrate is SiO2 and the cover layer is air). To achieve the high diffraction coefficient for the 1st-order transmission light, the parameters of the SiO2/air grating are chosen as period a 2 = 2a, thickness h 2 = 1.4a and filling factor f 2 = 0.4. Figure 4 shows the spectra of the zero-order, 1st-order and 2nd-order transmission diffraction coefficient’s norm of the SiO2/air grating at normal incidence from the top air for TE polarization. When the normalized frequency of incoming light is within the high-reflectance band [0.572, 0.653], the norm of zero-order transmission diffraction coefficient ranges from 0.427 to 0.325, while that of the 1st-order transmission diffraction coefficient changes from 0.552 to 0.571. At normal incidence, the diffraction coefficient of the mth-order equals to that of the –mth order. The wave numbers of the 1st-order and −1st-order diffraction light satisfy the inequality (1). So, in the high-reflectance band, the SiO2/air grating can efficiently diffract the normal incidence TE polarization into the 1st-order with the diffraction angle from 31.6° to 36.8° in the SiO2 and −1st-order with the diffraction angle from −36.8° to −31.6° in the SiO2.
Fig. 4 Spectra of the zero-order, 1st-order and 2nd-order transmission diffraction coefficient’s norm of the SiO2/air grating (period a 2 = 2a, thickness h 2 = 1.4a, filling factor f 2 = 0.4) at normal incidence from the top air for TE polarization.
At last, we discuss the OWDG (shown in Fig. 1) by cascading the subwavelength Si/SiO2 grating and the SiO2/air diffraction grating. Based on the results above, we choose the parameters of the non-symmetric gratings as period a 2 = 2a, thickness h = 0.3a, h 2 = 1.4a and filling factors f = 0.36, f 2 = 0.4. There are still two parameters undetermined yet. Those are the relative displacement between Si/SiO2 grating and SiO2/air grating and the thickness d of the homogeneous SiO2 layer. As to the relative displacement, we consider two typical structures, invariant under the mirror reflection σx that changes x to –x, are denoted by OWDG-A and OWDG-B respectively in Fig. 5 .
Fig. 5 Two typical structures of OWDG, denoted by OWDG-A and OWDG-B, are designed to be invariant under the mirror reflection σx that changes x to –x.
To reduce the coupling between the Si/SiO2 grating and the SiO2/air grating, the thickness of the homogeneous SiO2 layer is chosen to be 4a. Figure 6(a) shows the transmittance spectra of the OWDG-A and OWDG-B from the top air (denoted by T c) at normal incidence for TE polarization. While, Fig. 6(b) shows those from the bottom SiO2 (denoted by T s) at normal incidence for the same polarization. The transmittance from the substrate is the same for OWDG A and B, because the transmittance from the SiO2 substrate of both structures is mainly zero order transmission diffraction which is insensitive to the relative displacement between Si/SiO2 grating and SiO2/air grating. On the other hand, the transmittance from the air cover of OWDG A and B is mainly ± 1 orders transmission diffraction of SiO2/air grating, which is influenced by the relative displacement between Si/SiO2 grating and SiO2/air grating. Simulation results in Fig. 6 prove that the unidirectional transmission of OWDG-A is better than that of OWDG-B. So, we focus on OWDG-A.
Fig. 6 Transmittance spectra of the OWDG-A and OWDG-B (thickness d = 4a) at normal incidence for TE polarization from the air cover (a) and the SiO2 substrate (b).
To compare the unidirectional transmission of OWDG-A with different thickness d of the homogeneous SiO2 layer, we present the simulation results of the transmittance spectra of the OWDG-A with different d at normal incidence for TE polarization from the air cover in Fig. 7(a) . And, Fig. 7(b) shows the spectra of the ratio of two transmittances denoted by T C /T S. It is not hard to find that in the high-reflectance band of Si/SiO2 grating [0.572, 0.653], T C /T S is mostly greater than 100, and the transmittances T C is greater than 0.58 for d equals to 0.5a, a, 2a and 4a. Especially, for the OWDG-A with d equal to 4a, the transmittances T C is greater than 0.65, T C /T S is greater than 340 and T S is less than 0.0022 when the normalized frequency of normal incident TE polarization belongs to [0.581, 0.645]. So, a broadband unidirectional transmission of light for a large range of the thickness d can be realized with the proposed OWDG.
Fig. 7 (a) Transmittance spectra of the OWDG-A at normal incidence for TE polarization from the air cover and (b) the spectra of TC / TS with different thickness d of the homogeneous SiO2 layer
In order to investigate the angular dependence of the OWDG, Fig. 8(a) and 8(b) show the spectra of the transmittance T C (a) and T C /T S (b) of the OWDG-A (thickness d = 4a) for the TE polarization with the incident angle θ equals to 0°, 2°, 5° and 10°. Comparing with Fig. 3, we can see a bandwidth of unidirectional transmission of the OWDG decrease with the increment of the incident angle, and the invalidity of the Si/SiO2 grating as a reflector results in the failure of the unidirectional transmission of OWDG-A when the incident angle θ increases to 5°.
Fig. 8 Transmittance spectra (a) and the spectra of TC / TS (b) of the OWDG-A (thickness d = 4a) for the TE polarization with the different incident angle.
4. Conclusion
In order to realize unidirectional transmission, we have proposed a novel OWDG made with isotropic material by cascading a subwavelength grating and diffraction grating with a homogeneous spacing layer. Taking the fabrication of the gratings into consideration, the OWDG constituted with Si and SiO2 are investigated by using scattering matrix method based on plane-wave expansion. It has been demonstrated that more than 65 percent of the incident-wave energy can be transmitted unidirectionally with less than 0.22 percent transmission in the opposite direction at normal incidence for TE polarization. And, the relative bandwidth of the unidirectional transmission is greater than 10 percent.
References and links
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