Numerical and experimental demonstration of inverse designed low-index polarization-insensitive wavelength demultiplexer

The computational inverse design has paved the way for the design of highly efficient, compact, and novel nanophotonic structures beyond human intuition and trial-and-error approaches. Apparently, with nanophotonic design power, the exploration and implementation of multi-objective, complex, and functional nanophotonic devices become feasible. Herein, we used a recently emerged inverse design framework to demonstrate the design of a 1 × 2 polarization-insensitive wavelength division multiplexer (PIWDM) made of a low-refractive-index material with an index of 1.55. The inversely designed PIWDM structure successfully steers toward the targeted channels for 1.30 µm and 1.55 µm with TE and TM polarizations. Taking advantage of the design with a low refractive index material, we scaled the structural dimensions corresponding to the microwave region, fabricated the compact device using a 3D printer, and conducted an experiment as a proof of concept. The transmission values of the fabricated PIWDM device were −4.87 and −2.18 dB for TE and −2.19 and −2.23 dB for TM polarization at WG-I and WG-II, respectively.


Introduction
During the last two decades, researchers have tried to merge nanophotonics applications with optical communication technologies to design reliable, compact, and multifunctional * Authors to whom any correspondence should be addressed. devices [1]. Frequently encountered optical interconnection devices in optical communication systems include steering devices such as power splitters [2], mode division demultiplexers [3], and specifically wavelength demultiplexers (WDMs) [4]. Conveying different wavelengths to specific waveguides can be considered a great opportunity to redesign new compact and efficient optical communication systems. To develop a WDM, two different approaches are available as passive and active demultiplexing, the first of which is based on prisms, diffraction gratings, and spectral (frequency) filtering devices, and the other is based on a combination of passive components and tunable detectors, where each detector is tuned to a specific frequency [5]. Here, the preferable method for designing integrated WDMs is using a passive demultiplexing approach, which is feasible in terms of cost and power efficiency. In this regard, the design of passive and low-loss demultiplexing devices with compactness and multifunctional characteristics is a challenging issue, and its solution is widely demanding. In this direction, in recent decades, WDMs have attracted significant attention with state-of-the-art nanophotonic design tools and approaches [6][7][8][9][10][11][12][13][14].
In general, two types of design approaches for nanophotonic devices have been shown. The first method is a conventional educated intuition-based approach. With this method, the designers need to practice their skills regarding a parameter search and analytical formulation or incorporate search algorithms to reach the targeted device [6][7][8][9][10][11]. One example of these conventional methods is integrating a Mach-Zehnder interferometer (MZIs) and Si 3 N 4 on a silicon platform for the design of an on-chip polarization-insensitive WDM (PIWDM) device with a footprint of 2.5 mm × 0.9 mm [11]. The device was able to split four different wavelengths at quite wider footprints. The other approach is based on advanced algorithms and combined simulations to seek a solution that minimizes (or maximizes) a single objective or multiobjective related to the desired nanophotonics functionalities, also known as an 'Inverse Design'. Because non-heuristic methods provide an effective optimization of all structural parameters that are impossible with conventional methods, they have been used effectively in the design of nanophotonic devices [12][13][14][15][16][17][18][19][20][21]. A compelling example is that Huang and colleagues published a study of a WDM device using gradientbased algorithm [21]. Their blueprint was combining adiabatic waveguide tapers and on-chip WDM in the footprint of 2.4 µm × 10 µm. The device is able to separate incident light which is in fundamental mode and has only two different wavelengths, vertically. They exploited inversely designed adiabatic waveguides to minimize undesired mode orders as an input waveguide.
Optically conveying a light source with minimum loss is crucial in the design of efficient and integrable WDMs. Here, the important parameter of light guiding structures is their sensitivity to optical scattering losses resulting from input/output surfaces. In this regard, the application of low-indexcontrast (LIC) optical structures/waveguides made by polymers can be considered an effective solution for efficiently guiding a light source [22][23][24][25]. Moreover, low-index materials have great opportunities for low-cost and high-speed complex geometry fabrication to reveal optimization based (inverse designed) photonic devices. With the development of 3D printers, and stable, high-end, pure low-index materials, the production of the photonic devices is getting more reliable and preferable. In the light of this manner, there are also numerous low-index photonic technology studies in the literature. For instance, Estakhri et al revealed inverse-designed equation solver [26] which uses light to solve differential equations. To fabricate this device, they used polystyrene, which has only 1.52 refractive index value. The other compelling example is the inverse design of stretchable metalens study which is revealed by F Callewaert et al In this study, inverse-designed, disjoint, 3D printed low-index device have been tested in the microwave region. On top of that, they readily rearrange the distance of the parts of the device and this allow them to obtain new focal points [27]. Finally, E Bayati et al highlighted the importance of refractive index over the metasurface's performance. In this demonstration, they tested refractive indexes between 1.25 and 3.5 on the metasurface design using inverse design and forward design methodology. They claimed that low-index materials pave the way for 3D printing technologies, and this may reduce the time and cost necessary for these technologies [28].
We present the design and analysis of a three-dimensional (3D) WDM device, which operates at both transversemagnetic (TM) and transverse-electric (TE) polarizations and is realized using inverse design method. Because of the different interaction nature of the TE and TM polarized light with the designed metastructure, the combination of connected and isolated complex-shaped irregular gratings offers a smart design approach to independently manipulate the polarization state of the on-chip light propagation. We showed that the device splits two specific wavelengths which are 1.30 µm and 1.55 µm, since these wavelengths are already in use in optical communication systems independent of polarization with high efficiency. We scaled the structure up to adapt it to the microwave region and fabricated it using fused deposition modeling (FDM) with a polylactide (PLA) thermoplastic polyester material. Finally, we experimentally demonstrated its polarization-insensitive wavelength separation capability.

Methods and simulations
In this study, we used the open source Stanford Photonic INverse design Software (SPINS-b) [29] computational gradient-based optimization framework. We modified the design constraints and objective function to design a PIWDM structure. The framework aims to produce a fabricable nanophotonic device layout by receiving a user-defined objective function (performance criteria) as the input design parameters. It does not start with computations, but it first produces an image of the problem by setting up a 'problem graph'. The problem graph is made up of so-called nodes, which are the building blocks of the problem. These nodes are connected to each other in such a way that the product of a function feeds another function. To compute the final node-objective function input by the designer, for the problem graph, the algorithm starts a series of transformations of an initial randomly generated structure that is surrounded by the design area defined by the designer. These transformations contain continuous and discrete optimizations and a final discretization process. Although continuous and discrete optimization stages are subsequently implemented, the discretization process at the end forces fabrication constraints for the device. To achieve a final device, SPINS-b applies iterative convergence to reach the objective. Each iteration is a transformation, and each transformation calculates the problem graph. To compute the nodes in the problem graph for each transformation, the framework uses its own implementation of the finitedifference frequency-domain (FDFD) method for the optimization of the light matter interaction of the optical device. In short, the optimization method is combined with the FDFD method to achieve an optimization task defined by an objective function. Here, the objective function is an optimization problem defined by the designer. In our case, polarizationinsensitive wavelength demultiplexing is defined as an optimization problem and is defined analytically as follows [29]: where ω i represent the frequencies at 1.30 µm (i = 1) and 1.55 µm (i = 2); E i and H i are the electric and magnetic fields for the frequency ω i ; J i is the excitation current density at ω i ; A represents the E and H fields; c i is the overlap vector for A i ; the top (i = 1) and bottom (i = 2) output waveguide such that |c 1 A 1 | 2 and |c 2 A 2 | 2 are the power intensities forming the fundamental mode of the upper (labeled WG-II) and lower (labeled WG-I) waveguide, respectively; and f(x, y) = (x − y) 2 . Here, the main objective is to efficiently transfer optical power to the target channel regardless of the polarization. The objective function is an equally weighted sum of four sub-objectives, which corresponds to maximizing the transmission through the top waveguide at 1.30 µm and the transmission through the bottom waveguide at 1.55 µm for both polarizations. In order not to force the objective function to become more complex, terms for reducing the crosstalk are not added to the objective function. Thus, the reduction in crosstalk was provided indirectly. It is important to note that during the optimization process, each iteration was analyzed using 2D FDFD method, where the perfectly matched layer (PML) is used as the boundary condition. The optimization process was terminated at the 1200th iteration, and the final optimized structure was analyzed using 3D finite-difference time-domain (FDTD) method [30]. As stated before, in this study, we focus on the polarizationinsensitive wavelength demultiplexing effect. Figure 1 presents a visual representation of the demultiplexing schematic of the designed PIWDM, where it splits the incident broadband light (1.2-1.6 µm) into guided waves of two specified target wavelengths of 1.30 µm (channelized to the upper waveguide (WG-II)) and 1.55 µm (channelized to the lower waveguide (WG-I)) for both TE and TM polarizations. The output waveguides in the structure are located laterally to the input waveguide channel. The grid size 'a' defines the physical dimension of the unit cell and is fixed at 0.04 µm. The waveguide widths of the structure were selected as 0.80 µm (20a) and the input waveguide was positioned in the middle of the design area of 4 µm × 4 µm × 3 µm (100a × 100a × 75a) in the x-, y-, and z-directions, respectively. The centers of the output waveguides are located 1 µm above and below the middle of the design area. The device is designed for a light source with fundamental TE and TM polarizations with non-zero components of H z and E z in the z-direction, respectively. The computational domain is surrounded by perfectly matched layers (PML) to eliminate possible back-reflections. Because we intended to demonstrate the designed structure experimentally in the microwave region, the design material is selected to be PLA material with a low refractive index of 1.55. To validate the functional operation, 3D FDTD simulations were conducted for TE and TM polarized light, where numerical calculations were conducted only for the magnetic H z and electric E z fields, respectively. Figure 2 shows the operation of the designed PIWDM, where the demultiplexer device wavelengths of 1.30 µm and 1.55 µm into WG-II and WG-I, respectively, for both TE and TM polarizations. In addition, in the same figure, to provide the efficiency performance of the PIWDM structure, we calculated the transmission efficiency plots of the regarded channels within the operating wavelength range of 1.2-1.6 µm. It is important to note that the input channel is excited by both TE and TM polarized mode source.
In figures 2(a) and (b), the input source operating at a wavelength of 1.30 µm (a/λ = 0.0308, normalized frequency) shows signs of a strong confinement and guiding of light through WG-II. Figure 2 In addition, for the same channel WG-I, we can see that the electric field intensity in figure 2(e) has a slightly sharper rotation through it. Figure 2(f) shows the transmission values over a wide normalized frequency interval and at a/λ = 0.0258 for WG-I for TE and TM the numbers are -1.65 and -1.48 dB, respectively. The crosstalk values for WG-I are -8.94 dB for TE and -18.53 dB for TM polarization. It is important to note that the simulation was run under the high-performance computing and high-resolution meshing (a/4). The numerical results show that the most important objective of the inverse design optimization is accomplished, that is, a polarizationinsensitive wavelength-division is formed without a significant loss for both TE and TM polarizations. Moreover, the complexity of the structure produced by the algorithm is clear, and there is no doubt that such a structure cannot be designed intuitively.

Experimental demonstration of the inversely designed PIWDM structure
Based on normalized frequency values in section 2, the design can be applied to the desired spectrum region. Microwave experiments were conducted to demonstrate the operational principle of the PIWDM between normalized frequency The grid size is transformed from a = 0.04 µm to a = 0.80 mm which means that 20 000 times magnified structure. For a = 0.80 mm and a/λ = 0.0308, operating frequency becomes 11.54 GHz, corresponding with 1.30 µm in optical wavelength spectrum. The operating frequency related with wavelength of 1.55 µm becomes 9.68 GHz according to (a/λ = 0.0258) and a = 0.80 mm. The inversely designed PIWDM structure is fabricated using a 'Creality Ender 3 Pro' 3D printer, which uses the PLA material 'ESUN PLA + Silver' as a filament. PLA is a low-cost, biodegradable, and all-dielectric material with a low refractive index of n PLA ∼ =1.55 at 10 GHz. Moreover, a 3D printer is arranged for printing with a 100% infill ratio to obtain the solid and homogenous distribution of the PLA. Figure 3(a) shows a schematic illustration of the experimental setup used to elucidate the wavelength division performance of the PIWDM and its transmission efficiencies. Both horn and monopole antennas are employed to transmit and receive Gaussian profiled electromagnetic waves through Agilent E5071C ENA vector network analyzer (VNA), respectively. During the experiments, the xy-plane (scanning area) was scanned to obtain the electric field and magnetic field intensity distributions |E z | 2 and |H z | 2 , respectively, by arranging the appropriate orientations of the horn and monopole antennas for both TM and TE polarizations, respectively. In addition, another horn antenna, which is identical to the transmitter antenna, is utilized to measure the transmission values at the end of the output waveguides (WG-I and WG-II) for each polarization. A photographic illustration of the 3D-printed PIWDM and the constructed waveguides along with their physical dimensions are presented in figure 3(b). The width and the length of the fabricated device emerged as 8 cm, whereas the thickness was 6 cm. Furthermore, input and output waveguides are constructed using hollow rectangular aluminum foil coated cases to couple the light within the microwave regime. Moreover, these cases are considered to eliminate the absorption losses by ensuring that undesired reflections inside the cases are eliminated owing to the flat coating of the foils and guide and detect the wave at the input and output ports. All waveguides are almost identical and have spatial dimensions equal to 4 cm (length), 1.60 cm (width), and 6 cm (height) for the x-, y-, and z-directions, respectively.
For field intensity measurements, the monopole antenna was moved by 2-mm steps along both the x-and y-axes, and a precise interpolation was then applied to obtain the representative results. Similar to the numerical measurements in the simulations, the level of the z-axis is set to the middle of the structure. Figures 4(a) and (b) show the field intensity distributions at 9.68 GHz (a/λ = 0.0258) for the TM and TE polarizations, respectively. The longitudinal normalized crosssectional profiles are superimposed to clearly illustrate the intensity differences at the end of the output waveguides. Similarly, the electric and magnetic field intensity distributions are shown in figures 4(d) and (e) for TM and TE polarizations at 11.54 GHz (a/λ = 0.0308), respectively, with their normalized cross-sectional profiles. All field distributions clearly illustrate the successful separation of both 9.68 and 11.54 GHz frequencies at the end of the output waveguides.
Two identical horn antennas were used to obtain the transmission spectrums of the PIWDM structure between 9 and 12 GHz. One of the horn antennas is used to inject microwave as in the previous experiment and the other one is employed to measure the output power at the end of the outputs WG-I and WG-II. Input powers were measured at the end of input waveguide to obtain normalize transmission values for both output waveguides and polarizations. Figures 4(c)   experimentally measured values of the transmission graphs show the overall functionality of the PIWDM. It is important to note that the possible reasons for the discrepancy between numerical and experimental results include the impurity of the material, the production capability of the 3D printer, and external factors such as imperfect alignment. In addition, the experiment has a larger coupling loss because of the free space to the dielectric structure coupling loss, owing to the impedance mismatch.
The utilized computational algorithm produces a complex and compact metastructure with an ability to arbitrarily control of the light's polarization. The close inspection of the generated dielectric index distribution indicates the presence of both connected and isolated material parts as the objective includes the manipulation of both TE and TM polarizations, which have different oscillating planes for the electric and magnetic field components. Besides, the continuity of the tangential component of the electric field gives rise to different interactions as the polarization changes. Proof-of-concept of the designed device verified by experimental measurements shows that inverse designed PIWDM can efficiently separate wavelengths into different waveguide channels independently of their polarization state. Final structure exhibits high transmission ratio, low crosstalk values, compactness, and most importantly the independence of polarization, which can bring an important diversity to optical designs. In this regard, besides removing the necessary search for a polarizer, the device could also prove its main function of demultiplexing at 1.30 µm and 1.55 µm with great performance for both TE and TM polarizations. Moreover, all listed characteristics of the PIWDM revealed under a low refractive index of 1.55 by which normally the light is expected to have a lower performance because of weak light matter interaction behavior. As a result, the design satisfies the requirements of the deployed multi-objective function with a non-intuitive refractive index configuration.

Conclusion
In conclusion, we numerically studied a low-index dielectric medium capable of spectral demultiplexing for both TE and TM polarized light at 1.30 µm and 1.55 µm. We used a powerful design tool to independently manipulate TE and TM polarized light. Polarization-controlled low-index devices can find many applications in optical communications and photonic integrated circuits where in-plane photonic devices are essential parts of such applications. 1 × 2 PIWDM was designed using a low-index dielectric material (PLA) with the inverse design approach, and the transmission values are between −0.99 and −1.8 dB for both polarizations. To test the simulated device, it was produced using 3D printing technology and the experiment was carried out within the microwave region. Due to the low-index nature of the design, light coupling from external laser sources is expected to have low insertion losses as compared to high-index dielectric (such as Silicon) based devices. As a result of the experiment, the transmission values were −4.87 and −2.18 dB at WG-II and WG-I waveguides for TE polarization, respectively. Similarly, the transmission values were −2.19 and −2.23 dB at WG-II and WG-I waveguides for TM polarization, respectively.
Consequently, the experimental results support the design purpose of polarization-independent wavelength separation behavior. These results can pave the way for the inverse design of compact, high-efficiency devices that are polarizationindependent, with multiple output ports. With additional purposes added to the design algorithm, the structure can be achieved even with higher efficiency, a larger number of output ports, and even a narrower channel spacing. With the development of production techniques for asymmetric structures, it is thought that such devices can be easily used in commercial products. This might be a breakthrough for optical communications schemes that are widely dependent on polarization control. The PIWDM design is endowing a huge flexibility to the performance improvements of integrated photonic circuits, which may lead the way to several other different designs that may have more compact area constraints.

Data availability statement
No new data were created or analysed in this study.