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Abstract
A major challenge in inverse design of optical splitters is to efficiently reach platform nonspecific designs constrained to multiple functional requirements: arbitrary splitting ratio, low insertion loss, broad bandwidth and small footprint. While the traditional designs fail to fulfill all these requirements, the more successful nanophotonic inverse designs require substantial time and energy resources per device. Here, we present an efficient inverse design algorithm that provides universal designs of splitters compliant with all above constraints. To demonstrate the capabilities of our method, we design splitters with various splitting ratios and fabricate 1 × N power splitters in a borosilicate platform by direct laser writing. The splitters show zero loss within the experimental error, competitive imbalance of $\rm <0.5$ dB and broad bandwidth in the range 20 − 60 nm around 640 nm. Remarkably, the splitters can be tuned to achieve different splitting ratios. We further demonstrate scaling of the splitter footprint and apply the universal design to silicon nitride and silicon-on-insulator platforms to achieve 1 × 5 splitters with the footprints as small as 3.3 µm × 8 µm and 2.5 µm × 10.3 µm, respectively. Owing to the universality and speed of the design algorithm (several minutes on a standard PC) our approach renders 100 greater throughput than nanophotonic inverse design.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical splitters are indispensable components of classical and quantum photonic integrated circuits [1,2]. They combine the light inserted in an input port to produce a targeted intensity distribution at the output. While the $1\times 2$ optical splitters are used routinely, the rapidly growing need for space-division multiplexing (SDM) in optical transmission [3], computing [4] and sensing networks [5] translates into the need for multiport splitters.
Traditional design of multiports relies on repeated application of directional light coupling or multimode interference. Concatenation and nesting of $2\times 2$ directional couplers (DCs) offers continuous choice of splitting ratios (SR), simplicity of construction, and entanglement generation if both input ports are used [6,7]. However, the radiation losses at waveguide bends accumulate into a formidable insertion loss, which imposes the lower limit on the curvature radii and hence the splitter footprint [8]. DC operation at the quadrature point severely limits the output bandwidth [9]. Similarly, the loss in multiports obtained by concatenation of Y junctions scales with the number of junctions [10]. On the other hand, monolithic design of multimode interference (MMI) splitters responds well to the small-footprint requirements [11]. $1\times 2$ MMI splitters have a broad nearly-flat spectral response, but the dependence of the coupling length on wavelength shrinks the bandwidth in inverse proportion to the number of ports $N$ and thus impedes wide use of $N>8$ splitters [12]. Exploitation of MMI multiports has been further limited by the losses caused by the incomplete mode overlap and back reflections at the transition from the bulky central multimode to the output waveguides [13], as well as the difficulty to achieve arbitrary splitting ratios [12,14,15].
More recently, nanotechnology has enabled direct inverse design and realization of splitters constrained by multiple functional requirements [16,18]. These splitters are achieved by shaping the refractive index profile or meta-pattern on nanoscale by electron-beam lithography. For example, a metasplitter with a <1dB insertion loss, 40 nm bandwidth and ${\rm 3}.6 \mu m \times {\rm 3}.6 \mu m$ footprint has been designed in 48 h with moderate computing power [18]. While such high compliance with functional requirements is hardly accessible to traditional splitter designs, the exigent amounts of time and energy needed for design of a single platform-specific splitter are hardly acceptable for cost-effective large-scale manufacturing.
Here, we present an efficient inverse design algorithm and experimental realization of multiport splitters based on linearly coupled waveguide arrays (WGAs), which fulfill all above requirements. Namely, the absence of waveguide bends ensures negligible radiation losses, while the exclusive use of single-mode waveguides eliminates outcoupling losses and back reflections. In addition to the primary SDM capability, the flat spectral bandwidth admits $\rm >100$ of 100 GHz spaced WDM channels and propagation of ultrashort pulses. The key to the novel WGA-splitter design is exploitation of self-imaging, which allows for analytical intervention that dramatically accelerates the inverse-design algorithm. The robustness and potential of the method are demonstrated by successful splitter implementation in a borosilicate platform by femtosecond laser writing.
2. Inverse design
Under the conditions of coupled mode theory or tight-binding [19] and negligible material losses, a linearly coupled one-dimensional (1D) WGA is well-described by a tridiagonal real matrix $\mathbf {C}$. This matrix is composed of the coupling coefficients $\mathbf {C}_{j,j+1}$, such that $\mathbf {C}_{j,j+1}=\mathbf {C}_{j+1,j}$ for $j=1,2\ldots M$, and phase detunings between waveguide modes $\mathbf {C}_{j,j}$, where $M$ is the number (finite) of waveguides in the array [20]. Propagation of the light state $\vec {\psi }(z)=(\psi _1(z),\ldots \psi _M(z))^T$, represented by complex mode amplitudes $\psi _j(z)$, is modeled by the linear Schrödinger equation
with the stationary solution We further assume that all waveguides are identical and their modes phase matched: $\mathbf {C}_{j,j}=0$, which reduces the problem dimension to the number of independent coupling coefficients $M-1$. An example schematics is given in Fig. 1.
Fig. 1. A schematics of an integrated WGA splitter. The input and output ports extend from the main splitter body. Arrows show the light propagation direction. Ratios of coupling coefficients $\mathbf {C}_{j,j+1}$ are ubiquitous to all physical implementations. Interwaveguide separations $d_{j,j+1}$ are implementation specific.
Design of a WGA power splitter with the known input and output light intensities, $|\psi (0)|^2$ and $|\psi (L)|^2$, respectively, is a non-trivial multiparameter inverse problem, with non-unique solutions. Analytical solutions are available in a small number of cases which assume mirror symmetry of the array, $\mathbf {C}_{j+1,j}=\mathbf {C}_{M-(j-1),M-j}$ [15,21]. In order to efficiently find inverse solutions without imposing symmetry constraints on the coupling matrix, we limit the consideration to the WGAs that support light propagation with high degree of order, in particular periodic propagation. Periodicity occurs when all system eigenfrequencies are commensurate; i.e., when the ratio of any two eigenfrequencies is a rational number. Our previous analytic solutions to the inverse eigenvalue problem [22] enable fast calculation and optimization of the splitter output. We illustrate the design algorithm in Fig. 2 and describe its steps as follows:
Universal coupling matrix design algorithm
- • Choose eigenfrequencies and a free parameter space.
- • Express the coupling coefficients $\mathbf {C}_{j,j+1}$ as functions of eigenfrequencies as in [22].
- • Calculate the output wavefunction, Eq. (2), across the parameter space.
- • Evaluate the match of the wavefunction with the targeted splitter output for all free parameters.
- • If the match is good, give the corresponding coupling matrix as the output, otherwise explore a greater parameter space or different eigenfrequencies.
- • Select the waveguide refractive index profile.
- • Find the coupling coefficient dependence on waveguide separation as in [23].
- • Chose the nominal separation and calculate others based on coupling coefficients.
Fig. 2. Design algorithm. The universal algorithm accepts commensurate eigenvalues (EVs) and free parameters as entries and gives the coupling matrix $\mathbf {C}$ at the output. The implementation-specific algorithm takes $\mathbf {C}$, waveguide refractive index profile and the nominal separation $d$ as entries and produces the final coupler design. Dotted line shows an optional adaptive refinement of the search space.
We have tested several functions for evaluation of the output matching and found that minimizing the mean square error, $MSE=\sum _{j=1}^M(|\psi _j^{in}|^2-|\psi _j^{out}|^2)^2/M$, accurately selects splitters with predefined SRs. As the universal model is wavelength independent, we do not directly optimise splitter bandwidth. We use the well-known correspondence between the wavelength and coupling length – the longer the wavelength, the stronger the evanescent coupling and the shorter the coupling length, to select the broadband splitters. The positive selection rules are collocation of centroids and flatness of the light propagation patterns in all non-zero output ports. The pattern is considered flat if its FWHM is greater than 10% of the propagation period (revival length). Once the compliance with all set requirements is reached, the universal design is completed.
The implementation step requires adjustment of the interwaveguide separations to achieve the required ratios of coupling coefficients according to the formula
where $\alpha$ describes the decay rate of coupling constant with the interwaveguide separation. This parameter depends on the waveguide refractive index profile and is obtained numerically from the period of energy transfer between two coupled waveguides [21,23,24]. Thus, for a chosen waveguide profile, we first calculate $\alpha$ and then apply it to derive the geometry of the array. This completes the design procedure.Further, we analyze the algorithm efficiency. Given the eigenspectrum, the number of free optimization parameters is determined by the WGA symmetry. The eigenspectrum is defined by $(M-1)/2$ eigenfrequencies for odd $M$ and $M/2$ for even $M$. In a mirror symmetric array, the number of independent coupling coefficients equals the number of eigenfrequencies, leaving no free optimization parameters besides the coupling length. This severely limits the number of SRs accessible per eigenspectrum and requires exploration of a number of eigenspectra. On the other hand, an asymmetric WGA features $(M+1)/2$ free parameters for odd $M$ and $M/2$ for even $M$, which provides access to a variety of SRs and rarely requires exploration of different eigenspectra. This allows for fast convergence towards a satisfactory solution for coupling matrix, which takes several minutes on an average PC (e.g., we used the Intel Core i7-1065G7 CPU, 1.30GHz, 16 GB RAM). The lengthiest part of the implementation algorithm is the calculation of waveguide coupling parameter $\alpha$ which requires simulations of waveguide pairs. We have successfully benchmarked both 3D and 2D finite-difference beam-propagation simulations to the experimentally obtained results and were able to complete the implementation algorithm in several minutes. Therefore, conservatively, the proposed design procedure takes 5-10 minutes to obtain the universal coupling matrix and 10 minutes more to apply it to a particular waveguide setting. This is more than 100 times faster than the inverse design of meta splitters [18] and about 10 times faster than the inverse design of nanoshaped splitters [16], both implemented with more computing power than used here. We further stress that a single run of the universal WGA algorithm serves designs of all splitter instances with different waveguide index profiles, nominal separations and fabrication platforms, thus further increasing the algorithm throughput by an order of magnitude at least.
To demonstrate the capability of the proposed design procedure we produce a range of universal designs of equal and unequal splitters with one or more input ports and show them in Fig. 3. The splitter length is marked by a dashed vertical line, however the propagation is extended to the full self-imaging cycle to illustrate the periodicity of propagation. For example, the array in Fig. 3(a) serves as 1 $\times$ 5 splitter at the propagation length 0.44 $\pi$ and as 1 $\times$ 2 splitter at the propagation length 0.75 $\pi$, while the propagation period is completed at 1.5$\pi$. Here, the central waveguide is used as an input port. Remarkably, the same coupling coefficients can be used to construct other splitters providing that different input ports and coupler lengths are used. The 1$\times$4 splitter from Fig. 3(c) uses the first waveguide WG1 as the input port, while the 2$\times$4 splitter in Fig. 3(d) uses waveguides WG1 and WG2 as input ports. Due to the mirror symmetry of the array around the central waveguide, the same results are obtained when the waveguides WG5 or a pair of waveguides WG4 and WG5 are used as input ports instead. Predefined splitting ratios are achieved with excellent accuracy that predicts minimal loss to other waveguides. Due to the splitter length - wavelength correspondence, slow dependence of the SR on the splitter length, indicates broad bandwidth. This is the case with the equal $1\times 2$ and $1\times 3$ splitters in Fig. 3(a) and (b) and the unequal splitters in Fig. 3(e) and (f), which display flat light propagation patterns at the splitter output. The flatness is a consequence of constructive interference and resembles the band flatness achieved in broadband MMI splitters. On the other hand, $1\times 4$ and $1\times 5$ equal splitters operate in quadrature and resemble narrowband DCs.
Fig. 3. Light propagation through splitters based on WGAs with 5 waveguides. Splitter begins at $z=0$ and ends with dashed line. The propagation is extended beyond the splitter length to show its periodicity. (a) Equal $1\times 2$ and $1\times 5$ power splitters with $\mathbf {C}_{1,2}=\mathbf {C}_{4,5}=1$, $\mathbf {C}_{2,3}=\mathbf {C}_{3,4}=0.6236$. (b) Equal $1\times 3$ splitter with $\mathbf {C}_{1,2}=1$, $\mathbf {C}_{2,3}=0.7489$, $\mathbf {C}_{3,4}=0.6916$, $\mathbf {C}_{4,5}=0.4634$. (c) Equal $1\times 4$ splitter with $\mathbf {C}$ as in (a). (d) Equal $2\times 4$ splitter with $\mathbf {C}$ as in (a). (e) Unequal $1\times 3$ splitter with SR 1:0:2:0:3 and $\mathbf {C}_{1,2}=\mathbf {C}_{4,5}=1$, $\mathbf {C}_{2,3}=\mathbf {C}_{3,4}=\sqrt {3/2}$. (f) Unequal $1\times 3$ splitter with SR 2:0:3:0:2 and $\mathbf {C}_{1,2}=1$, $\mathbf {C}_{2,3}=2.4143$, $\mathbf {C}_{3,4}=2.0289$, $\mathbf {C}_{4,5}=2.5797$.
3. Experimental results and discussion
3.1 Fabrication
The splitters were fabricated by direct femtosecond laser writing [26,27], as it is sketched in Fig. 4(a). Ultrashort pulses from an ATSEVA ANTAUS Yb-doped fiber laser, at a repetition rate of $500$ kHz and a central wavelength of $1030$ nm, are focused on an $L=5$ cm-long borosilicate glass wafer (refractive index $n_0=1.48$), with a thickness of $1.1$ mm and a width of $10$ mm. The fs laser pulses modify the material properties at the focusing region, with a refractive index change of $\Delta n\approx 10^{-4}-10^{-3}$. Straight waveguides are created by slowly translating, at a velocity of $0.4$ mm/s, the glass wafer along the propagation coordinate $z$ by means of a motorized Aerotech XYZ stage. At a writing power of $\sim 130$ mW, each fabricated waveguide is single mode at $640$ nm and larger wavelengths [24].
The experimental characterization of WGAs is performed by using the setup sketched in Fig. 4(b), which is based on a Supercontinuum (SC) YSL SC-5 laser source emitting in the range $450-2400$ nm. An acousto-optic filter reduces the operative wavelength range to $450-1450$ nm, with a resolution of $5$ nm and $\sim 1$ mW for each wavelength. The laser beam is aligned and polarized and, then, focused into the photonic chip (PC) by means of a microscope objective (MO). The output profile is collected by a second MO and imaged onto a CCD camera. An example in Fig. 4(c) shows a $1\times 4$ splitter upon excitation at $740$ nm.
Fig. 4. (a) Femtosecond laser writing technique. (b) Characterization setup. (c) Example of a 1$\times$4 splitter at $740$ nm. (d) Z-scan method, where $z_i$ describes different lengths for the shorter waveguides, while the longer one is used for excitation (see arrow). (e) White light microscopy of a couple of dimers and trimers for coupling characterization. (f) White light microscopy of two five-port splitters.
The characterization of coupling constants is typically performed by fabricating a set of splitters having one full ($L$) and one short ($z_0$) waveguide, in order to catch the coupling dynamics before the first oscillation cycle [24]. Every splitter is fabricated having a different waveguide separation and, after extracting the power ratio between both waveguides, the coupling constant is obtained by:
An exponential fit of the form $A\exp (-\alpha d)$ is applied and the expected coupling constants can be inferred. This method is good enough for fabricating larger systems as standard photonic lattices [28–30], where a dynamical scale (more than an exact and precise value) is required. However, this method is not as exact as necessary for the present work, where the coefficients require a higher precision. Considering this, we implemented a more precise calibration method to obtain the coupling constants, which is based on a $z$-scan method [24]. By implementing this, we obtain the dynamics along the propagation coordinate $z$ at different positions in the glass wafer, therefore capturing also some glass inhomogeneities across the wafer. This is certainly a more precise method from a statistical point of view and, also, it allows us to match the expected dynamics from model (1) with the experiments. This can be performed by using any waveguide configuration; in the present work, we implemented this by using dimers and asymmetric trimers as sketched in the Fig. 4(d). We show a couple of examples in Fig. 4(e). After this calibration is performed and more precise coupling constants are extracted, we proceeded to fabrication of equal power splitter configurations, as the examples shown in Fig. 4(f).To realize splitters from Fig. 3(a)–(c), we first applied implementation-specific algorithm with the experimentally determined coupling constant to obtain the exact interwaveguide separations and then used them as input for the fabrication procedure. In addition, we numerically modelled the light propagation through the splitters. We directly compare the numerical and experimental results in Fig. 5 and observe an almost perfect agreement in between the predicted output and the measured experimental image. Not to occult real information, all the experimental images are shown in a linear color scale. As a result, some traces of light are visible in waveguides designed to carry zero power. This kind of “parasite” intensities are within our measurement error and could be simply erased by implementing a splitter configuration as the one sketched in Fig. 1.
Fig. 5. Numerical design and experimental realisation of WGA equal power splitters from Fig. 3 (a)–(c) by femtosecond laser writing in borosilicate. Light propagation is simulated for 1D array by the finite-difference beam propagation method for experimentally obtained step-index waveguides with ${\rm n}_{co}=1.4804$, ${\rm n}_{sub}=1.48$, and width 4.4 $\mu$m. We set the interwaveguide distance ${\rm d}_{1,2}=16 \mu m$ and calculate the other distances by Eq. (3). The simulations were performed at the wavelength of 633 nm.
The splitters were characterized in the wavelength range $580-750$ nm as shown in Figs. 6 and 7. This range is determined by the guiding properties of the written waveguides, which finds its optimum around $650$ nm for the specific material and fabrication parameters. The corresponding performance parameters of splitters are given in Table 1. In what follows, we discuss these results and compare them to the splitters reported in literature.
Fig. 6. Spectral response of five-port splitters with performance: (a) $1\times 2$, (b) $1\times 3$, (c) $1\times 4$, and (d) $1\times 5$.
Fig. 7. Multifunctionality. (a) A splitter performs either as $1\times 2$ or $1\times 4$ equal splitter at different wavelengths. (b) A WGA renders $1\times 5$ or $1\times 2$ splitting at different lengths.
Table 1. Performance parameters of fabricated WGA splitters
3.2 Insertion loss and imbalance
We first determine typical performance parameters for equal power splitters – the total insertion loss ${\rm IL}$ and the power imbalance ${\rm IB}$. Due to the singlemodedness of all waveguides and the possibility to extend the constitute splitter waveguides into input and output ports, as shown in Fig. 1, the incoupling, outcoupling and back-reflection losses can be neglected. Hence, the principal source of loss is leak into the ports designed to carry zero power. We define the total insertion loss as the total power transmitted to all non-zero output ports
Remarkably, we achieve zero loss within the experimental error. It is apparent in Fig. 6 that for all splitters the nominally zero ports carry no power across a respectable bandwidth. We describe these “ideal” splitters by the bandwidth $\Delta \lambda _{\rm IL=0}$ and minimum imbalance $\rm \{IB_{IL=0}\}_{min}$ in the zero-loss operation mode. The results are given in the last two columns of Table 1. For completeness and easier comparison with the results of others, we also report the insertion loss averaged over 0.5 dB bandwidth, ${\rm \langle} IL\rangle _\lambda$.Imbalance is a parameter specific to equal splitters and is defined [11] as the ratio of the maximum to minimum power in non-zero output ports
The imbalance averaged over 0.5 dB bandwidth, ${\rm \langle} IB\rangle _\lambda$, remains below 0.5 dB for all splitters except $1\times 5$, for which it reaches 1.1 dB. “Ideal” single-wavelength splitters have almost perfectly balanced splitting ratios.We observe that WGA splitters are significantly less lossy than the transmission-optimized arc-shaped $1\times 4$ Y-junction splitter (IL=0.48 dB, IB=0.09 dB) [10], low-loss MMI splitters ($1\times 2$ splitter IL=0.1dB and $1\times 12$ splitter IL=1.13 dB, IB=0.72 dB) [13,31–33], nanoshaped $1\times 3$ splitter (IL=0.64 dB, IB=0.64 dB) [16] and $1\times 4$ meta splitter with significant scattering loss on air holes (IL=1.4 dB) [18]. From the above comparison, we also conclude that all designs perform well-balanced splitting.
3.3 Bandwidth
The splitter bandwidth is estimated as a spectral range in which the intensity in all output ports does not deviate by more than 0.5 dB from the design value. Such definition takes into account both the insertion loss and the imbalance. For example, for $1\times 2$ splitter in Fig. 6(a) the loss to two nominally zero waveguides limits the bandwidth despite an almost zero imbalance, whereas for $1\times 5$ splitter in Fig. 6(d) the imbalance between all ports limits the bandwidth despite the by-definition zero total insertion loss.
Further analysis of results in Fig. 6 shows that the flattened light propagation profiles of $1\times 2$ and $1\times 3$ splitters render bandwidths of 60 nm and 20 nm centered at 635 nm and 650 nm, respectively. A simple estimate shows that they are capable of guiding ultrashort femtosecond pulses and accommodating $\rm >300$ and $>100$ GHz spaced WDM channels, respectively. On the other hand, bandwidths of $1\times 4$ and $1\times 5$ splitters centered at 687 nm and 611 nm, respectively, do not surpass 5 nm. These results corroborate the design rules and the correspondence of spectra with the propagation dynamics discussed in section 2. Availability of analytical solutions enables comparison of the ideal 0.5 dB bandwidths of symmetric 1 $\times$ 2 WGA, MMI and DC splitters. All relative bandwidths group around 10% and drop dramatically with the number of ports. On the other hand, the relative bandwidths of the reported $1\times 4$ meta splitter (1.5%) [18] and $1\times 3$ nanoshaped splitter (4.2%) [16] exceed 0.7% and 3% bandwidths of the respective WGA splitters. While the WGA splitter bandwidth was not optimised here, this is possible by one of the strategies applied to DCs, namely waveguide asymmetrisation [34] and curving [35], or by insertion of subwavelength gratings [36,37]. However, this significantly complicates design and fabrication. A more efficient strategy may be to increase the free-parameter space by oversizing the array, as indicated by our results for broadband $1\times 2$ and $1\times 3$ splitters produced by 5-waveguide arrays. It will be the subject of future work.
3.4 Multifunctionality
Dependence of the SR on wavelength and splitter length is further exploited to build multifunctional splitters. In the measurement shown in Fig. 7(a), the SC input laser wavelength was tuned to find that the same WGA can be used as a broadband $1\times 2$ equal splitter centered at 640 nm and a narrower band $1\times 4$ equal splitter centered at 690 nm. Changing the length of a WGA while keeping the wavelength fixed has the analogous effect. For example, a 7.5 mm long array in Fig. 7(b) functions as $1\times 5$ splitter and a 16.5 mm long array as a $1\times 2$ splitter.
In both examples, we designed the coupling matrix for a specific SR and then observed the light propagation dynamics through that WGA in search for another useful SR, at a different length. Such search is likely to give a positive result in the WGAs that support self-imaging, similarly to MMI splitters [11]. However, in general, setting two or more SR criteria simultaneously increases the number of requirements beyond the number of free design parameters. This overdetermined problem can be relaxed by permitting the use of different input ports for different splitter functionalities, as in Figs. 3 (a) and (c), or by phase tuning of the light inserted into different input ports [21,38].
Multifunctionality of WGA and MMI splitters allows for application of the same lithographic mask in production of different splitters and hence has potential to reduce costs of their high-end manufacturing.
3.5 Footprint scalability
Scaling up of the circuit capacity by increasing the number of ports stands in opposition to the small-footprint requirement. The straightforward mitigation by reducing the spacing between optical waveguides leads to the evanescent coupling between them, so called optical crosstalk, that impedes further downscaling [39]. We use the fact that our splitters do not exclude crosstalk, but rely on it, to achieve favourable scaling of the splitter footprint. Reduction in interwaveguide separation decreases the splitter width linearly and the coupling length exponentially [24,27]. We have confirmed this by producing $1\times 5$ splitters with different nominal interwaveguide separations, as shown in Fig. 8.
Fig. 8. Scaling of the $1\times 5$ WGA splitter length with the minimum waveguide separation. Y-axis scale is logarithmic.
Nevertheless, the question of how far we can reduce the splitter size remains open. The basic design assumption, that of the periodic light propagation, does not impose any fundamental limits on splitter size. However, closely spaced waveguides evanescently couple beyond the nearest neighbours making it difficult to solve the inverse eigenvalue problem and construct the coupling matrix, which now features more than two side diagonals. This is the principal technical limitation of our approach and, at the same time, a promising research direction for future work on high-density optical integration, where not only the next-nearest neighbor interactions become relevant but also the coupled-mode approach becomes questionable and new numerical tools mandatory.
The limitation is less severe in platforms with high refractive index contrast which support stronger mode confinement and a still valid coupled-mode approach [40]. We checked this by 3D simulation of the $1\times 5$ WGA splitters with closely spaced waveguides in silicium nitride (SiN) (${\rm n}_{co}=2.02$, ${\rm n}_{sub}=1.453$ at 800 nm) and silicon on insulator (SOI) (${\rm n}_{co}=3.673$, ${\rm n}_{sub}=1.444$ at 1550 nm), Fig. 9. In SiN, we work with waveguide separations of 200 nm and obtain the splitter footprint of 3.3 $\mu$m $\times$ 8 $\mu$m, which is 6 times smaller than the footprint of $1\times 4$ MMI splitter in the same platform [41]. In SOI, we set the waveguide separation to 100 nm and obtain 2.5 $\mu$m $\times$ 10.3 $\mu$m $1\times 5$ WGA splitter, which is comparable to 3.6 $\mu$m $\times$ 3.6 $\mu$m 1 $\times$ 4 meta splitter [18] and 3.8 $\mu$m $\times$ 2.5 $\mu$m $1\times 3$ nanoshaped splitter [16], reported to be the smallest in literature. We note that the simulated semiconductor splitters have rectangular step-index profile which proves independence of the proposed universal design on waveguide geometry.
Fig. 9. Semiconductor couplers. (a) $1\times 5$ splitter from Fig. 3(a) in SiN platform with the waveguide width of 500 nm, height 400 nm and the minimum waveguide separation 200 nm. (b) Equivalent $1\times 5$ splitter in SOI platform with the waveguide width of 400 nm, height 300 nm and the minimum waveguide separation 100 nm.
Finally, we consider the impact of fabrication tolerances on SR. We performed extensive statistical estimates of the impact of deviations in the coupling coefficients and detunings from their design parameters and found that the former heavily dominates the overall error in SR. Using the experimentally derived exponential coupling coefficient dependence on the waveguide separation, we estimate that the tolerance of 50 nm in waveguide positioning is needed to keep the root mean square error (RMSE) of SR below 10%. This is achievable by high-precision translation stages, such as the Aerotech stage used in this work. On the other hand, the same RMSE can be maintained despite the excursions in the waveguide refractive index, which linearly propagate into the excursions in detunings, as large as 15 %.
4. Conclusion
We have proposed and demonstrated a new type of multiport integrated optical splitter with competitive low loss, bandwidth and footprint, that is entirely based on linearly coupled WGA. The inverse design of splitters is achieved by an efficient algorithm that assumes regular periodic light propagation without a priori restrictions on WGA layout nor the targeted splitting ratios. The required accuracy of the coupling coefficients is achieved through precise determination of the coupling constants by an adapted $z$-scan method. The corresponding fs-laser written equal power splitters show zero insertion loss at bandwidths of up to 20 nm, and $\rm <0.1$ dB loss at bandwidths of up to 60 nm, while maintaining low imbalance $\rm <0.5 dB$. These excellent performance parameters corroborate the design algorithm, while its universality and supreme speed put it ahead of competition. Furthermore, the periodicity of light propagation pattern enables delivery of multiple splitting ratios by a single WGA, thus generating potential for further savings in fabrication process. Besides the performance and economic advantages, the proposed design offers new possibilities for quantum optics, logical operations, sensing and communications. For example, the evanescent coupling between waveguides results in continuous quantum walk [42,43], thus enabling direct application of the demonstrated equal splitters in multiphase quantum interferometry and entangled W-state generation [38,44], while the multimode coupling demonstrated in fs-laser written WGAs [24] and the universality of the design procedure presented here open the door to new multi-mode splitter designs for mode-division multiplexing schemes. Finally, to further increase the capacity of these and the low-loss SDM links achievable by the proposed splitters, our future work will concentrate on general inverse algorithms that produce solutions for unlimited number of ports.
Funding
Fondo Nacional de Desarrollo Científico y Tecnológico (1191205); Millennium Science Initiative Program (ICN17_012); Ministry of Science, Technological Development and Innovation of the Republic of Serbia (451-03-47/2023-01/200017).
Acknowledgments
The authors acknowledge fruitful discussions with A. Maluckov, and support from Millennium Science Initiative Program, FONDECYT, and the Ministry of Education, Science, and Technological Development of the Republic of Serbia..
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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