1. Introduction

Manipulation of optical spatial modes in integrated photonic circuits has attracted much attention due to huge potential applications in mode-division multiplex, efficient waveguide coupling and all-optical logic devices [1,2]. One promising basic component is linear photonic crystal (PhC) mode converter that converts waveguide modes between different orders following reciprocity. Three expected key characteristics of mode converter are large operating bandwidth, high transmission/conversion efficiency and compact structure. In last decade, two mechanisms were utilized: phase matching between different waveguides [3,4] and scattering/coupling effects introduced by defect or cavity in single waveguide [5–8].The drawbacks of PhC mode converter based on latter scheme include narrow bandwidth due to resonance and difficulty in fabrication due to performance sensitivity of defect or cavity. However, from the perspective of reducing device size which is crucial for on-chip applications, the latter scheme prevails. V. Liu [7] proposed and numerically analyzed a two-way mode converter in rod-type PhC waveguide with a functional region occupies 4 × 10 unit cells of perfect PhC, which was most compact at the time. The unit cell of perfect PhC is the minimum periodical region containing the basis of PhC and will be referred as only “unit cell” for simplification in the rest of paper. Very recently, L. Frandsen [9] experimentally achieved a one-way PhC slab waveguide mode converter by topological optimizing air-holes within the functional region about 7 × 6 unit cells. The experiment gave a satisfactory verification of numerically design and provided a good example of fabricating PhC with complicated element shapes.

Conventionally, magneto-optical effect [10,11] and optical nonlinearity [12,13] are utilized to break reciprocity to obtain optical diode and isolator. Like electric diode, the optical diode processes the capability of unidirectional transmission of light. Unfortunately, common optoelectronic materials present only relatively weak magneto-optical effect, and the high operational optical intensity threshold of nonlinear effect would limit the on-chip usage as well. Passive and linear structures are highly desirable. Therefore, linear PhC mode converter becomes an important building block to achieve optical unidirectionality, based on the mode conversion in a spatial asymmetric manner while still keeping the reciprocity. It should be noted this kind of unidirectional transmission only holds for specific mode due to symmetry of scattering matrix. Nevertheless, such reciprocal device is still perceived as optical diode but not isolator. Isolator is similar but should block all possible modes in one direction. Recently, several schemes have been reported. X. Hu [5] and Z. Li [6] experimentally realized optical diode based on PhC heterojunction along −45°direction (diagonal direction), which resulted in directional bandgap mismatch and different mode transitions. The device sizes are about 13 × 13 unit cells and without explicit waveguide to reduce transmission loss. Besides mode converter, V. Liu [7] also numerically presented a optical diode within waveguide by introducing spatial asymmetry in 4 × 5 unit cells large functional region. The operating bandwidth is effectively enlarged after a further optimization on the radii of 20 rods. S. Feng [8] employed a rectangular defect between two perpendicular PhC waveguides which are specially designed to accommodate both even and odd mode. The operating bandwidth is very narrow due to resonance effect and forwarding transmittance is only about −6 dB.

Although efforts have been paid in designing linear PhC mode converter and optical diode for on-chip applications, large operating bandwidth, high efficiency, high unidirectionality, and especially small footprint of functional region still have plenty room for improvement. In this letter, we present designs of both mode converter and optical diode in linear rod-type PhC waveguide with only 4 × 1 unit cells large functional region, which are several time smaller than previous works and does not compromise key performances. More design freedoms are introduced in geometry optimization achieved by geometry projection method (GPM) [15] combining method of moving asymptotes (MMA) [16]. The optical field is simulated by finite element method (FEM).

2. Methodology

The two-dimensional square lattice PhC made of silicon rods immersed in air, with lattice constant a and radius of rod 0.2a, is commonly used in previous studies [3,7,8]. The relative permittivity of silicon is set ɛ = 12 for the near-infrared wavelength range without loss. The first photonic band gap for TM-polarized modes (electric vector parallel to rod) exists from frequency 0.281c/a to 0.417c/a, where c is light velocity in vacuum. As illustrated in Fig. 1(a), the proposed structure comprises a simple W2 waveguide which removes two rows of rods and a 4 × 1 unit cells large functional region which is specially optimized to manipulate the propagated mode. The perfect W2 waveguide can support fundamental even mode and first higher order odd mode, between which the mode conversion is desired in this paper. As an example, the spatial electric fields and power fluxes of specific mode at frequency 0.4c/a are shown respectively. Figure 1(b) demonstrates the dispersion curve for each mode calculated by scanning wave vector along x-axis in FEM. The lowest frequency of odd mode is 0.3526c/a at k_x = 0 point, indicating mode conversion is only available for higher frequencies.

 

Fig. 1 (a) Schematic diagram of proposed two-dimensional PhC mode converter and optical diode . PhC is based on silicon rods (ɛ = 12) in air. (b) Dispersion curves of even mode and odd mode supported by W2 waveguide. (c) Schematic diagram of GPM: The rod shape is determined by intersection line of level plane and three dimensional surface formed by control points. FEM + MMA are adopted to adjust heights of control points to find the optimum solution.

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Pursuing extremely compact functional region in waveguide for mode conversion and optical diode effect behooves us to take more freedoms into consideration, especially shape of specific PhC rods. Here, we adopt GPM to describe the dielectric distribution inside the functional region consisting of only 4 × 1 unit cells, as demonstrated in Fig. 1(c). Two dielectric rods inside the functional region are intentionally kept because same amount of rods would appears in perfect W2 waveguide. This choice can also avoid large amount of trivial scattered rods. The GPM employs 6 × 20 control points whose heights can be fitted as a three-dimensional surface with refined mesh (40 × 140 in this case). The mesh choice is a balance between computational cost and flexibility. The intersection curve of this surface with a pre-defined level plane is used to define the interface between different dielectric materials in two dimensions. When level plane is set at zero, the dielectric constant at coordinate x can be expressed as [17] $\epsilon (x)={\epsilon }_{air}+\frac{({\epsilon }_{si}-{\epsilon }_{air})}{2}(\tan h[\frac{sign[S(x)]\cdot d(x)}{\xi }]+1)$, where S(x) is surface function, d(x) is minimum distance from the refined surface to the nearest intersection curve, and ξ is control parameter of vanishing of intermediate dielectric at interface. The essence of GPM is to control the rod shape, size and position by adjusting the heights of the control points. The data of dielectric distribution inside functional region from GPM is saved in the format that can be recognized by FEM. We reconstruct whole PhC structure as shown in Fig. 1(a): 13 arrays long x-axis with the functional region in which the data is imported. In vertical direction, 9 rows along y-axis are adopted at each side of W2 waveguide. The optical field and transmission properties are simulated by FEM. The maximum element size is below a/100 in the functional region and a/10 in other domains. The total number of degrees of freedom is kept larger than 5 × 10⁵.

In this work, we adopt MMA [16], an efficient gradient based optimization algorithm, to determine the heights of all control points in GPM for mode conversion and optical diode effect respectively. The objective functions are built based on the integrals of power flux of targeted optical mode in area region P_i as marked in Fig. 1(a). Sensitivities required by MMA are calculated using adjoint method based on FEM simulation [15]. Integrals of power flux in area is unconventional but a technical choice to obtain stable sensitivities for optimization. The initial condition comprises two circular rods with 0.2a radius which are located exactly as the perfect W2 waveguide. In order to extent the operating bandwidth of our designs, two optimization steps are taken: first, we initially search a candidate at center frequency 0.40c/a, next, further optimization is done based on the new objective function which is the weighted average of objective function values at five frequencies within the targeted frequency domain. The weights are adjusted to obtain a reasonable parabola-like transmission efficiency curve and good performance at center frequency. This optimization strategy ensures that mode conversion and optical diode effect can be achieved at all five frequencies, leading to operating bandwidth enlarged. The optimization is considered converged when the change in objective function is less than 0.0001 between iterations. Finally, the optimized shape of rods inside the functional region are approximated by polygons which are equally divided into 60 sides. The experimental fabrication would benefit from the polygons which avoid burrs and provide exact coordinates of smooth edges.

3. Results and discussions

3.1. Design of bidirectional mode converter

A bidirectional mode converter are designed to convert the incident fundamental even mode to the higher order odd mode in both forward and backward directions. Since the material of linear PhC here are described by only scalar permittivity and permeability which result in a symmetric scattering matrix, reciprocity should hold, implying that opposite type mode conversion, i.e. odd mode to even mode, can be intrinsically achieved as well. To enforce the symmetry between forward and backward direction, we introduce mirror symmetry inside the functional region as an additional optimization constraint. The objective function at single frequency in MMA can be expressed as

Figure 2(a) demonstrates the optimized rods inside functional region for mode converter. Two polygons are referenced by circles with radius r₀ = 0.3a and detailed normalized deviations between r and r₀, expressed as d = (r-r₀)/r₀, are shown in Table 1. It can be seen that huge deformations of circular rods achieve bidirectional even-to-odd mode conversion by introducing more geometric design freedom. The representative electric fields in Fig. 2(b) show satisfactory converted mode profiles in W2 waveguide. Quantitatively, the transmission efficiency ranges from about 60% to 82% within the targeted frequency domain from 0.39c/a to 0.41c/a. The maximum transmission efficiency is 82% at frequency 0.402c/a. If center frequency 0.40c/a corresponds to wavelength 1550 nm, the operating bandwidth is about 77.5nm which is much larger compared with previous full circular rod design [7]. It should be noted that the transmission is evaluated by line integral of power flux at output port. When optimization is performed at specific frequencies, the rod shapes vary with each other and the device commonly has a narrow bandwidth. Therefore, our design presented here reflects the compromise between efficiency and bandwidth due to the nature of scattering effects, and meanwhile keeps high quality of mode conversion at center frequency.

 

Fig. 2 (a) Optimized shape of rods inside functional region for broadband mode converter. (b) Spectrum of transmission efficiency and corresponding electric field profiles.

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Table 1. Deviations between optimized rods for mode converter and reference circular rods

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3.2. Design of optical diode

Let us move on to the design of optical diode. When different waveguide modes convert in a spatial asymmetric manner, the optical diode effect can be realized for specific mode based on linear PhC structure. Since the reciprocity is hold, such device is not optical isolator. Our purpose here is that even mode can be converted to odd mode and be transmitted in forward direction, but reflected in backward direction. Due to the definition of optical diode, there should be no symmetry in the functional region, unlike the case of mode converter. The objective function for broadband optical diode, based on two FEM model whose even mode incidence is at left and right port respectively, can be expressed as

Figure 3(a) demonstrates the optimized rods inside functional region for optical diode. Two polygons are referenced by circles with radius r₀ = 0.3a and detailed normalized deviations between r and r₀, expressed as d = (r-r₀)/r₀, are shown in Table 2. The spatial symmetry of rods is obviously broken. The field profiles, demonstrated in the insets of Fig. 3(b), explicitly show the asymmetry of the transmission of particular even mode between forward and backward directions. Quantitatively, the unidirectionality ranges from 19dB to 30dB within the targeted frequency domain from 0.395c/a to 0.405c/a. For center frequency 0.4c/a, the forward transmission efficiency is 83% while backward value is 0.081%, resulting in approximate 30 dB unidirectionality. If center frequency 0.40c/a corresponds to wavelength 1550 nm, the operating bandwidth is 38.7 nm. The performances of presented optical diode, including forward transmission efficiency, operating bandwidth, and unidirectionality are comparable to or even better with previous designs [3,7,8]. However, the functional region occupies only 4 × 1 unit cells, which is about two wavelengths wide and half wavelengths long. To the best of our knowledge, these are the most compact designs of linear dielectric mode converter and optical diode at the present time.

 

Fig. 3 (a) Optimized shape of rods inside functional region for broadband optical diode. (b) Spectrum of unidirectionality and corresponding electric field profiles.

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Table 2. Deviations between optimized rods for optical diode with reference circular rods

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Next, we consider the tolerance performance of the proposed designs. $\pm \text{5\%}$and $\pm \text{3\%}$fluctuations are randomly added to the polygons with respect to each r of perfect optimized rods for mode converter and optical diode respectively. The random fluctuation is used to simulate the imperfections which may be generated in experimental fabrication. Three imperfect samples are examined for each design. The shapes of imperfect sample #1 are illustrated in the inset of Fig. 4(a) and 4(b) as an example. It can be seen that fluctuation introduces both dense burrs and slight shape deformations. No obvious performances degradation have been observed, indicating a good robustness of proposed designs. In some occasions, the performance seems even better. However, this better performance is unpredictable and should not be taken as an advantage to proposed designs because such random dense burrs cannot be fully reflected in GPM and break some of the criteria (e.g. geometric symmetry and weight of frequencies) in optimization process. Moreover, it is worth noting that the performance of optical diode is more sensitive to the rod shape. Although the change of absolute value of backward transmission efficiency is small, but the relative variation is quite large compared with forward transmission efficiency, resulting in the higher sensitivity of unidirectionality. This is the reason for lower tolerance and lower operating bandwidth of optical diode design. Finally, we have verified the functionality of both mode converter and optical diode designs in three dimensional models by finite-difference time-domain. Such real rod-type PhC device will naturally suffers radiation loss in direction parallel to rods due to lack of confinement, which will reduce the transmission efficiency.

 

Fig. 4 (a) Spectra of transmission efficiency of mode converters with perfect and 5% imperfect shape. (b) Spectra of unidirectionality of optical diode with perfect and 3% imperfect shapes.

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4. Conclusions

In summary, designs of extremely compact broadband mode converter and optical diode based on linear rod-type photonic crystal have been proposed. The functional region consists of only 4 × 1 unit cells in the W2 waveguide. The dielectric distribution inside functional region are numerically optimized by FEM combining GPM and MMA. We have clarified the key performances including transmission efficiency, operating bandwidth and unidirectionality. Although the length of functional region is only about half vacuum wavelength, the performances are not compromised compared with previous works. Moreover, the reasonable tolerance of proposed designs have been demonstrated. We expect the heuristic compact designs can be applied as potential building blocks in future all-optical integrated circuits.