We applied nonlinear optimization to design integrated digital metamaterials in silicon for unidirectional energy flow. Two devices, one for each polarization state, were designed, fabricated, and characterized. Both devices offer comparable or higher transmission efficiencies and extinction ratios, are easier to fabricate, exhibit larger bandwidths and are more tolerant to fabrication errors, when compared to alternatives. Furthermore, each device footprint is only 3μm × 3μm, which is the smallest optical diode ever reported. To illustrate the versatility of digital metamaterials, we also designed a polarization-independent optical diode.
© 2015 Optical Society of America
Devices that enable asymmetric light transmission, that is, light transmits in one direction, but doesn’t transmit in the opposite direction, are extremely useful for a variety of photonic applications. However, their implementation in integrated devices is challenging . This functionality can be achieved by breaking the Lorentz symmetry condition, typically via various nonlinearities. Examples of such non-reciprocal devices include those that use magneto-optic materials [2–6 ], metamaterials  or indirect interband photonic transitions . Devices that utilize combinations of metal and dielectric or magneto-optical materials are not compatible with CMOS fabrication technologies. Furthermore, such non-reciprocal devices tend to be large and require significant power input. Although non-reciprocal devices are necessary for certain applications such as optical isolation, asymmetric light transmission (or optical diode behavior) can be achieved with much simpler passive devices via spatial symmetry breaking as long as the functionality is limited to a finite number of input modes . Such devices are simpler to fabricate and are significantly smaller than their non-reciprocal counterparts. Examples of such devices include those based on metamaterials , photonic crystals [11–14 ], metallic-silicon waveguide , and ring resonators . Recently, an ultra-compact optical diode (size ~3μm × 5μm) that utilizes a combination of photonic crystal and gratings was proposed . However, this device operates only for TM polarization and for frequencies outside the telecommunications band. Achieving polarization insensitive optical diodes is challenging and most of the devices presented previously are polarization dependent. This is due to the fact that photonic crystals can typically manipulate only one polarization state. By combining waveguides with meta-surfaces, a polarization independent optical diode was recently demonstrated in the microwave regime . This device is not readily extended to the telecommunications band due to the complexity of fabrication.
Here, we overcome previous limitations by applying the concept of digital metamaterials [17,18 ], that is, devices where the local permittivity is engineered in a fabrication-constrained fashion in order to achieve desired mode-conversion functionality. Specifically, we designed, fabricated and characterized integrated, all dielectric (silicon on silicon-dioxide) optical diodes each with dimension of 3μm × 3μm, which is the smallest such device ever reported. We experimentally verified the performance of two such devices, one for each polarization state. We also designed a third device, which exhibits polarization independent diode behavior at the expense of slightly higher insertion loss. Digital metamaterials is a subset of free-form metamaterials, where mode conversion is achieved by optimized 3D or 2D nanophotonic geometries [19,20 ]. By imposing fabrication constraints, we ensure that the devices are not only CMOS compatible and easy to fabricate, but are also robust to fabrication errors.
Each device is comprised of 30 × 30 square “pixels” of size 100nm × 100nm, resulting in a total area of 3μm × 3μm. Light enters and exits the device via 3μm-wide multi-mode waveguides as illustrated in Fig. 1 . Polarized light launched from left to right propagates through the device, while that launched in the opposite direction is reflected. All devices are designed for λ0 = 1550nm and are fabricated on a silicon-on-insulator (SOI) substrate with top silicon thickness of 300nm and an oxide thickness of 3μm. The device for TE polarization is illustrated in Fig. 1(a) and that for TM polarization is illustrated in Fig. 1(d). The white regions represent air, where the silicon has been etched away (black regions represent unetched silicon) and the etch depth is the same as the depth of the waveguide, 300nm. This further implies that the device can be fabricated at the same time as the waveguides, and a separate lithography step is not required. For TE, the simulated forward (from left to right) transmission efficiency is 71.1% and the backward efficiency is 1.8%. While for TM, the simulated forward and backward transmission efficiencies are 91.1% and 3.2%, respectively. These efficiencies are comparable to those reported for previous devices, but our devices are significantly smaller. The simulated steady-state intensity distributions for both devices are shown in Figs. 1(b)-1(c) and Figs. 1(e)-1(f) for TE and TM polarizations, respectively. The field distributions indicate that mode conversion is primarily enabled by coupled guided-mode resonances in the nanostructures. The refractive index variation introduced by the metamaterial device creates a perturbation of the input mode. As a result, multiple guided-mode resonances are excited so as to create a transmission band for the forward propagation direction but a forbidden band in the backward direction.
Digital metamaterials may be designed using a variety of optimization algorithms. Here, we applied a relatively simple variation of the direct-binary-search (DBS) algorithm. DBS was previously used to design broadband non-imaging optics [21,22 ], free-space polarizers , integrated devices , and nanophotonic light-trapping structures [23,24 ]. Here, we further adapt the algorithm to design the digital metamaterials illustrated in Figs. 1(a) and 1(d). The 3μm × 3μm device is first discretized into 30 × 30 pixels and each pixel could exist in two possible states: silicon or air. Different pixel-state distributions will exhibit different permittivity distributions and thereby, distinct electromagnetic properties. DBS is then implemented to optimize the pixel-state distribution that gives us the desired electromagnetic properties. Here, DBS tried to increase the forward transmission efficiency, while minimizing the backward one. The search algorithm was described previously in ref . All the 900 binary pixels are traversed in random order within each iteration. Proper termination conditions such as a minimum improvement in FOM and maximum iteration numbers are imposed to guarantee numerical convergence. Because of the algorithm’s tendency of premature convergence to local maxima, we repeated the same optimization process with several randomly generated initial candidates, among which the best optimized solution was chosen. In addition, we parallelized the algorithm and used Amazon’s EC2 service to expedite the optimization. Using one micro cluster composed of three virtual machines, each with 32 virtual CPUs, the optimization time was ~270 hours. An open-source finite-difference time domain (FDTD) solver (MEEP) was used to simulate the full 3D distribution of electromagnetic fields within our designs .
Although the devices illustrated in Fig. 1 could be fabricated via a single-lithography step, we adopted a two-step process, since we do not have access to high-resolution optical-projection lithography. Heidelberg MicroPG 101 (laser pattern generator) was first used to pattern 3μm-wide waveguide, and the Oxford 100 reactive-ion etcher (RIE) with a gas mixture of SF6 and C4F8 was used to etch the silicon. The second step, which defines the metamaterial diode used the dual-beam focused-ion-beam (FIB, FEI, Helios 650) system. Fiducial marks were used to enable alignment between the two steps. Other details of the fabrication process was described in . The scanning-electron micrographs of the final devices are shown in Figs. 2(a) and 2(b) .
The measurement system is sketched in Fig. 2(c)  and measurement steps were similar to the ones described in . The polarization controllers (PC1 and PC2) were first calibrated using the on-chip polarizer. The on-chip polarizer consists of a straight waveguide with a vertical air slot near the center of the waveguide. The center of the air slot has a 70nm offset with respect to the center of the waveguide. The structures as well as the field pattern for the on-chip polarizer are summarized in Fig. 3 . For the on-chip polarizer, TM is transmitted efficiently, while TE is blocked as shown in Figs. 3(c) and 3(d). The on-chip polarizer allows us to control the input mode (TE or TM). The alignment between the output polarization plane and the polarizer was achieved by adjusting PC2. The polarization components of the output light could be selected by rotating the polarizer accordingly.
In order to rule out the impact from butt-coupling loss between the lensed fiber and the waveguide as well as the propagation loss of silicon waveguide, a straight 3μm-wide waveguide without any patterns was used as a reference. Normalizing the measured transmission efficiency for the waveguide containing the optical diode against that for the unpatterned waveguide provides the measured transmission efficiency for our metamaterial device. The measured forward and backward transmission efficiencies for both devices are shown in Figs. 2(d) and 2(e). The measured spectra are close to the simulated ones. Small differences between them are probably due to the edge roughness of the waveguide and small errors in alignment among the 2 lithography steps. We measured forward and backward transmission efficiencies of 62.1% and 2.8% for TE, and 79.8% and 10.4% for TM.
For true optical isolation, the backward transmission efficiency for modes of any order should be zero . This can be achieved only by devices that break Lorentz symmetry (reciprocity). Magneto-optical or optical non-linear materials are the most common choices for such non-reciprocal devices. However, as stated earlier, such materials are not compatible with CMOS technology, and these devices tend to be much larger and require power. On the other hand, passive optical diodes are reciprocal devices and as such cannot be optical isolators in the general sense. Their diode function is only valid typically for a single input mode [10–13 ]. Our devices are also reciprocal and passive, but because of the generality of their design can exhibit diode functionalities for more than 1 input mode. We numerically investigated the performance our TE diode under illumination by higher order modes and the results are summarized in Fig. 4 . In Fig. 4(a), we plot the extinction ratio, defined as the ratio between forward and backward transmission efficiencies, as a function of the incident mode order. Since our device is designed for the fundamental mode, the highest extinction ratio is observed when the mode order is 1 and it deteriorates at higher orders. However, the deterioration for the second order mode is small and optical diode behavior is sufficiently preserved as illustrated by the steady-state light-intensity distributions in Figs. 4(c) and 4(d). When the same simulation is performed for the 3rd order mode (Fig. 4(e)), the diode behavior is not seen (see Figs. 4(f) and 4(g)). Thus, our device is an effective diode for the first two TE modes.
We numerically investigated the robustness of our designs to fabrication errors. Specifically, we varied the device (silicon) thickness that is determined by the etching depth of the pixels. The resulting forward and backward transmission efficiencies for the two devices are shown in Figs. 4(h) and 4(i), respectively. As expected, the efficiency drops as the thickness changes from the design value of 300nm. If we are able to tolerate an efficiency drop of 20% from the peak value, we can specify the allowable thickness variation as ± 26nm.
We also simulated the time evolution of the electric field within the two devices in order to visualize their performance. The resulting animations are included as supplementary information. Due to the subwavelength structures within each device, evanescent modes are excited. For the forward transmission, these evanescent modes constructively interfere and give rise to multiple resonant modes that are propagating in the plane of the device. When the TE diode is illuminated in the backward direction, the excited evanescent modes interfere destructively, leading to negligible light in the output waveguide (see Media 1). When the TM diode is illuminated in the backward direction, the evanescent modes interfere so as to excite resonant modes that are coupled out of instead of into the output waveguide (see Media 2). The asymmetry in the spatial distribution of refractive indices gives rise to the drastic difference in transmission in the two directions. In addition, we performed a 2D discrete Fourier transformation (DFT) of our nanophotonic structures (diode Ez) to gain further insight into the physics. For comparison, a similar nanophotonic structure but with random pattern is also analyzed. The results are summarized in Fig. 5 . The left column shows the permittivity distribution of the device and the right column shows the 2D DFT results. It is found that the random (Fig. 5(a)) pattern yields wavevectors that are too diffuse, which means that input energy is transferred to almost any supported mode, most of which cannot radiate efficiently into the output waveguide. However, for our optimized device, the wavevectors are substantially limited to the central area denoted by the red circle in Fig. 5(b), which correspond to modes that can radiate efficiently into the output waveguide. This means input energy could be efficiently transferred to the propagation modes in the output waveguide.
5. Polarization-independent optical diode
We can apply the concept of digital metamaterials to design a polarization-independent optical diode. The resulting device geometry is illustrated in Fig. 6(a) and the simulated steady-state light-intensity distributions for both polarization states at the design wavelength (1.55mm) are shown in Figs. 6(c)-6(f). The simulated forward and backward transmission efficiencies for the 2 polarization states are summarized in Fig. 6(b). Although the insertion loss is higher, the device operates as a reasonable diode for both polarizations. In particular, we simulated an extinction ratio of 8.9dB and 10.5dB for TM and TE, respectively. Our device is particularly interesting because it is thousands of times smaller than an alternative polarization independent device proposed recently . The simulated time evolution of the electric field within the polarization independent unidirectional transmission device is incorporated in the supplementary information as Media 3.
We designed, fabricated, and characterized ultra-compact integrated reciprocal optical diodes, devices that efficiently transmit light in one direction, while blocking it in the opposite direction. These devices are an example of digital metamaterials that enable fabrication-friendly, yet highly functional devices that are significantly smaller than alternatives. To the best of our knowledge, our optical diodes are the smallest such devices ever reported. We measured forward and backward transmission efficiencies of 62.1% and 2.8% for the TE diode and 79.8% and 10.4% for the TM diode, respectively. Furthermore, numerical studies indicate that the TE diode maintains its performance for the first 2 incident orders. Finally, we also designed a polarization-independent optical diode that is only 3μm × 3μm in size. It is important to point out that digital metamaterials can enable almost any linear mode-convertor, and can be optimized for functionality, size and ease of fabrication.
The authors thank Jose Dominguez-Caballero for assistance with the direct-binary-search algorithm and Brian Baker for the sample preparation. We thank Peng Wang for assistance with the waveguide fabrication and Steve Blair for use of the tunable fiber laser. This work made use of University of Utah shared facilities of the Micron Technology Foundation Inc. Microscopy Suite sponsored by the College of Engineering, Health Sciences Center, Office of the Vice President for Research, and the Utah Science Technology and Research (USTAR) initiative of the State of Utah. This work made use of University of Utah USTAR shared facilities supported, in part, by the MRSEC Program of the NSF under Award No. DMR-1121252. Financial support was provided by National Aeronautics and Space Administration (NASA) (NNX14AB13G), U.S. Department of Energy (DOE) (EE0005959), and University of Utah.
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