1. Introduction

The optical diode has attracted huge interests for its capability of unidirectional light propagation, which invokes potential applications in integrated optical circuits for all-optical computing and information processing [1–4]. Highly desirable key performances of optical diode are high unidirectionality, low insertion loss, large operational bandwidth and small device footprint. Two types (reciprocal and non-reciprocal) optical diodes have been proposed based on different mechanisms. The conventional non-reciprocal designs utilize either magneto-optical effect [5–7] or nonlinearity [8–10]. The reciprocity of the Lorentz theorem is broken in such designs and more importantly the optical isolator, which hinders the propagation of any possible mode in one direction, can be achieved [11]. The major limits include relatively low magneto-optical coefficient in silicon-based waveguide materials, the integration issue caused by external magnetic field and relatively high optical power required by the nonlinearity. Therefore, reciprocal optical diodes, in which the time-reversal inversion symmetry holds, have drawn increasing attention. The underlying mechanism is the spatial asymmetric mode conversion in linear structures [11,12], resulting in that unidirectional transmission naturally exists for only certain modes.

In the last decade, due to the flexibility of tuning photonic band gap and designing coupling structures, the linear two-dimensional (2D) photonic crystal (PhC) has been shown as building blocks to realize the reciprocal optical diode [13–21]. Both narrowband and broadband designs have been demonstrated for 2D rod-type PhC. For narrowband, the mode coupling between cavity and two adjacent waveguides was utilized [13,14]. For broadband, the proposed schemes included 1) phase matching between split waveguides [15], 2) 90degree bend and directional coupler [16], 3) presence and radii optimization of rods inside functional region [17], and 4) deformed rods inside functional region [18]. However, above works based on rod-type PhC were confined in theory and simulation because of the difficulties in fabrication and the absence of light confinement along the rod direction as well. Compared with rod-type PhC, the fabrication technique of photonic devices based on 2D air-hole PhC slab is more mature and even compatible with conventional CMOS processing, but lack schemes for optical diode can be found. Derived from the directional bandgap mismatch and different mode transition [19], X. Hu [20] and Z. Li [21,22] experimentally realized the optical diode effect in air-hole PhC slab with heterojunction. The peak of forward transmission efficiency, the operational bandwidth and the maximum forward/backward contrast ratio were 25%, 40nm and 0.885 respectively [21,22]. More recently, L. Frandsen [23] for the first time fabricated an even-to-odd mode converter inside a PhC waveguide obtained from topology optimization. The functional region comprised 34 deformed air holes. The insertion loss reached as low as −2 dB within 43nm bandwidth. This optimization did not take the backward transmission into account (not an optical diode), but presented an excellent validation of numerical design with complicated element shapes in experiment. Still, the new compact broadband design for optical diode with high performance is highly desirable. In this paper, we present a reciprocal optical diode in linear air-hole PhC waveguide with 1.2a$\times$2.8a functional region, which possesses only one deformed air hole. The optical properties are simulated by 2D finite element method (FEM), and the air hole shape is obtained by the geometry projection method (GPM) combining the method of moving asymptotes (MMA).

2. Design

The fundamental structure is a 2D PhC constructed by air holes in a silicon slab arranged in a triangular lattice with lattice constant a. The radii of all air holes are r = 0.35a. The effective refractive indices of silicon and air are set 2.983 [23] and 1 for 2D simulation. The design of the multimode PhC waveguide is shown in Fig. 1(a). The waveguide core is formed by removing one row of air holes along x-direction and moving the first and second nearest row with 0.15a and 0.1a in y-direction respectively. Figure 1(b) illustrates the dispersion curves for even mode and odd mode of TE polarization (electric field parallel to the slab) calculated by scanning the wave vector along x-direction. One period PhC waveguide with periodic boundary condition is adopted in this eigenfrequency analysis. It can be observed that an overlap region in frequency exists for both modes which ensures the possibility of mode conversion. To more explicitly show the light propagation in designed PhC waveguide, the magnetic fields of even and odd mode at frequency 0.36c/a are simulated and plotted in Fig. 1(c). The corresponding transmission efficiencies are approximate 92.6% for even mode and 89.8% for odd mode. A conventional silicon waveguide of length L = 2a and width W = 1.6a is located at each side of the PhC waveguide to take the insertion loss induced by standard coupling into account. The operational frequency region of our optical diode is chosen from 0.355 c/a to 0.365 c/a shown as shadow region in Fig. 1(d), in which transmission efficiencies of both modes keep higher than 83%.

 

Fig. 1 (a) The design of the multimode PhC waveguide (b) Dispersion curves of TE polarization (c) Magnetic field H_z for even mode and odd mode propagated in designed PhC waveguide at f = 0.36c/a (d) Transmission efficiency of even mode (solid line) and odd mode (dashed line). The shadow region represents the chosen operational frequency region of optical diode.

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To achieve optical diode with high unidirectionality, low insertion loss, large operational bandwidth and especially small device footprint, it behooves us to consider new design freedom: the shape of air holes in the PhC waveguide. Figure 2(a) depicts the schematic structure of the proposed optical diode. The light propagation from left to right (right to left) is denoted as forward (backward) direction. The whole PhC domain has 11a in length (x-direction) and 11.4a in width (y-direction). At the center of PhC waveguide, the functional region, which is responsible for forward mode conversion and backward transmission blockade, occupies only 1.2a$\times$2.8a area. The GPM [24] is used to describe and control the air hole shape, size and position inside the functional region by tuning the heights of control points. In this calculation, 13$\times$29 control points and the refined 65$\times$145 meshes are adopted to generate the fitting three-dimensional(3D) surface, whose intersection line with the level plane zero defines the boundary of the air hole in functional region. Based on this boundary, the relative permittivity at each point can be determined by [25] $\epsilon (x)={\epsilon }_{air}+({\epsilon }_{si}-{\epsilon }_{air})(\tan h[sign[S(x)]\cdot d(x)\eta ]+1)\text{/2}$, where d(x) is the minimum distance to the intersection line, S(x) is the surface function and $\eta$controls the transition width of relative permittivity near the boundary. More details of GPM can be found in [24].

 

Fig. 2 (a) Schematic structure of the proposed optical diode, including two conventional silicon waveguides, a PhC waveguide and a functional region. The area of functional region is 1.2a$\times$2.8a. (b) The zoomed-in functional region with the obtained dielectric distribution. The boundaries of the air hole are plotted as solid lines (smoothed results) and circles (directly from GPM).

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For realizing the optical diode effect, we adopt the gradient based optimization algorithm MMA [26] to determine the heights of all control points in GPM. The objective functions are built by the integrals of power flux for targeted optical behavior. In each iteration, the spatial refractive index data in functional region from GPM is loaded in the FEM model to calculate the related optical properties and especially the sensitivities required by MMA [26]. More than 6$\times$10⁵ degrees of freedom in FEM is kept for accuracy. Here, the searching strategy for final broadband optical diode comprises three stages. Because the even-to-odd mode conversion in forward direction is the pre-requisite, in first stage our target is the bidirectional mode converter at the center frequency 0.36c/a. In this stage, a mirror symmetry along x-direction is induced inside the functional region, resulting in that only half control points (7$\times$29) need to be considered. The objective function is $F_{MC}(\omega )={\displaystyle \underset{P_1}{\int }P(\omega )ds+}{\displaystyle \underset{P_3}{\int }P(\omega )ds}-{\displaystyle \underset{P_2}{\int }P(\omega )ds}$, where P is the power flux in x-direction and P_i are the forward output areas shown in Fig. 2(a). The integral of power flux in areas is not the best but a technical choice to approximate odd mode output with stable sensitivities. The maximum objective function ensures high efficient mode conversion. In the second stage, based on the result of previous mode converter, the mirror symmetry is then broken and the backward transmission of even mode is added into the objective function: $F_{OD}(\omega )={\displaystyle \underset{P_1}{\int }P(\omega )ds+}{\displaystyle \underset{P_3}{\int }P(\omega )ds}-{\displaystyle \sum _{i=2,4,5,6}^{}{\displaystyle \underset{P_i}{\int }P(\omega )ds}}$. The maximum of this objective function ensures both high forward transmission with odd mode output and low backward transmission, meaning an optical diode working at single frequency. The last stage is to broaden the operational frequency range to 0.01c/a (from 0.355c/a to 0.365c/a). Weighted average of objective functions at five frequencies is used to ensure reasonable performance within the targeted frequency domain. It can be expressed as $F_{BOD}=\frac{\text{1}}{\text{5}}{\displaystyle \sum _{i=1}^5{w}_i{f}_{OD}({\omega }_i)}$, where w_i is the weight which can be adjusted to obtain parabola-like unidirectionality. The numbers of iterations for three stages are around 500, 100 and 200 respectively. Figure 2(b) demonstrates the final result for broadband optical diode. The solid smooth line is obtained by b-spline method based on GPM points (empty circles). Only one deformed air hole locates inside the functional region which certainly benefits the experimental fabrication compared with 34 deformed air holes in previous work [23]. It should be noted that present design cannot be considered as neither the strictly best nor the only, but a satisfactory solution for the functional region. Very recently, another objective-first inverse-design algorithm [27] has been adopted to achieve asymmetric spatial mode conversion in silicon waveguide by taking the magnetic field at the boundary as the objective, whereas we take the integrals of power flux in our calculations.

3. Results and discussions

The optical field and power flux of the proposed reciprocal optical diode in linear air-hole photonic crystal waveguide at center frequency 0.36c/a are shown in Fig. 3 as an explicit functional demonstration. Two exciting conditions are considered. In forward direction, the even mode excited at left port of the silicon waveguide couples into the PhC waveguide and transits to odd mode due to the deformed air hole in the functional region, and then transmits and couples to the output silicon waveguide. The total forward transmission efficiency is 91.4%. On the contrary, even mode excited at right port is blocked in the backward direction and nearly no light can be detected at the left output waveguide. The backward transmission efficiency is only about 0.39%. The unidirectionality, defined as 10log₁₀(T_for/T_back), reaches approximate 23.7 dB. The optical diode effect in present design derives from the spatial asymmetry inside the functional region. Since the PhC is considered linear, reciprocity of the Lorentz theorem holds. As a consequence, when odd mode is excited at right side, the odd mode transits to even mode backwardly with high efficiency and cannot be blocked, indicating our design is not an optical isolator.

 

Fig. 3 Simulated magnetic field H_z and power flux (x-direction) at center frequency 0.36c/a in (a) forward direction and (b) backward direction.

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The large operational bandwidth is one of the desired performances of an optical diode. If a cavity or a defect is introduced into the waveguide, a narrow pass band should be expected due to the nature of scattering [13,14]. In this work, the last stage of the searching strategy plays an important role to ensure the efficient forward mode conversion (low insertion loss) while keeping the low backward transmission within frequency range from 0.355c/a to 0.365c/a. The performances at two boundary frequencies are for certain degraded but acceptable in the design. The forward transmission efficiency keeps higher than 89.9% and the backward value keeps lower than 2.4% as shown in Fig. 4(a). The lowest backward transmission efficiency locates around the center frequency 0.36c/a. Moreover, we here demonstrate the quality of the even-to-odd mode conversion in the forward direction. The solid lines in Fig. 4(b) represent the power fluxes of input even mode and output odd mode at center frequency respectively. The two flux peaks of the odd mode are almost equal in height. The mode extinction ratio [23] is larger than 20dB, showing that our design is a high-quality mode converter. The results also validate the choice of the objective function. When the frequency leaves away from the center, it is obvious that the heights of two peaks slightly vary with each other but still keep the clear double-peak shape.

 

Fig. 4 (a) Transmission efficiency of forward and backward even mode propagation. (b) The output power flux of odd mode in forward direction.

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The parabola-like unidirectionality of the proposed optical diode keeps higher than 15 dB within the operational bandwidth 0.01c/a. The maximum value reaches approximate 24 dB near the center frequency. In our calculation, the GPM and MMA are always combined with FEM. Here, we also put the deformed air hole inside functional region with smooth lines into the 2D finite-difference time-domain(FDTD), which provides good consistent results as shown in Fig. 5. Moreover, an air-bridged 3D model is built in FDTD to validate the performance of the real device. In experiments, the air-bridged structure can be achieved by removing insulator layer (SiO₂) underneath the silicon PhC slab using HF solution [21,22]. The relative permittivity of the silicon slab is set 12. The curves of unidirectionality obtained with slab thickness 0.55a and 0.54a are plotted in Fig. 5. It can be seen that the curve simply shifts in frequency when varying the slab thickness. The maximum unidirectionality is about 19.6 dB while the value keeps higher than 12.7 dB for h = 0.54a, indicating an acceptable optical diode effect [21,27]. The degradation mainly comes from the reduction of the forward transmission efficiency, from 35% to 41% within the operational bandwidth in 3D simulation. The reduced transmission efficiency is still comparable with the reported mode converter design with standard butt coupling [23] and the optical diode design based on silicon waveguide [27].

 

Fig. 5 The unidirectionality in dB as a function of frequency calculated by FEM, 2D FDTD and 3D FDTD.

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Finally, the tolerance of the deformed air hole inside the functional region is taken into consideration. We set the center wavelength at 1550 nm, at which the corresponding PhC lattice constant a = 558 nm. As shown in Fig. 6(a), the perfect proposed design exhibits optical diode effect within 40 nm bandwidth (from 1530 nm to 1570 nm), which is at same level compared with all previous broadband works based on the 2D air-hole PhC [20–23]. Here, 10 nm in y-direction to the upper and lower boundary of the deformed air hole is considered as the maximum of experimental fabrication error in three situations: expanding(noted as + 10nm), shrinking(noted as −10nm) and random fluctuations. The imperfect samples and their unidirectionalities obtained from FEM simulation are plotted in Fig. 6(b) and 6(a) respectively. The changes include the shift of peak and the reduction of unidirectionality. No obvious degradation and device failure can be observed although size variation and random burrs are introduced, indicating a good robustness of the proposed design.

 

Fig. 6 (a) The unidirectionality of perfect and imperfect samples at 1550 nm center wavelength. (b) The upper and lower boundaries of the corresponding imperfect samples: expanding ( + 10nm), shrinking(−10nm), and random fluctuations #1 and #3.

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4. Conclusions

In summary, a design of compact broadband optical diode in linear 2D air-hole PhC waveguide is presented. The dielectric distribution inside functional region with area 1.2a$\times$2.8a is obtained by FEM combining GPM and MMA. Only one deformed air hole locates inside the functional region to achieve the optical diode effect within 0.01c/a bandwidth. The parabola-like unidirectionality ranges from 15 dB to 24 dB while forward transmission efficiency keeps higher than 89.9%. The optical properties of 3D air-bridged models and the tolerances of the design are also demonstrated and validated. The links between performance of 3D device and 2D optimization should be further investigated. We expect the proposed heuristic reciprocal optical diode can play an important role in all-optical integrated circuits.