Ideal, constant-loss nanophotonic mode converter using a Lagrangian approach

: Coupling light between an optical ﬁber and a silicon nanophotonic waveguide is a challenge facing the ﬁeld of silicon photonics to which various mode converters have been proposed. Inverted tapers stand out as a practical solution enabling efﬁcient and broadband mode conversion. Current design approaches often use linearly-shaped tapers and two dimensional approximations; however, these approaches have not been rigorously veriﬁed and there is not an overarching design framework to guide the design process. Here, using a Lagrangian formulation, we propose an original, constant-loss framework for designing shape-controlled photonic devices and apply this formalism to derive an ideal constant-loss taper (CLT). We speciﬁcally report on the experimental demonstration of a fabrication-tolerant, 15- µ m -long CLT coupler, that produces 0.56 dB ﬁber-chip coupling efﬁciency, the highest efﬁciency-per-length ratio ever reported.


Introduction
The large index of refraction contrast of a silicon-on-insulator (SOI) platform enables highdensity, cost-effective photonics on a chip. [1,2] The typical interconnecting channel waveguide on 220 nm SOI has a 450 nm width resulting in a tightly confined mode profile. This type of waveguide is often used to connect integrated devices together since it minimizes the mode area of the waveguide, thereby enabling high-density optical circuitry while keeping the propagation loss small over short distances. [3] However, at the junction between the interconnecting waveguide and the chip external coupling interface there is an intrinsic mode mismatch requiring mode conversion. [4][5][6][7][8][9][10][11][12] Tapers are used to gradually convert the mode area and the effective index of the guided mode, typically by varying the waveguide width. In Fig. 1(a), the effective mode area (EMA) of a 220 nm thick SOI waveguide is shown as a function of the waveguide width (w) using a full-vectorial mode solver (FULL, solid line) at λ = 1550 nm.
The top-left inset shows the waveguide geometry with the main regions of interest: core, side, and corner. It is observed that the EMA is minimized for the single-mode condition at point (c) (w = 450 nm, Fig. 1(a)). Here, the EMA is defined as: [13] EMA = S z dxdy 2 S 2 z dxdy (1) where S z = (E × H) ·ẑ is the Poynting vector,ẑ is the unitary vector along the waveguide's propagation axis. Between points (c)→(d), the EMA undergoes a direct, linear expansion as shown in the bottom inset of Fig. 1(a). During this process, the guided mode remains primarily confined in the silicon core ( Fig. 1(c) and 1(d); FULL) since its effective index remains substantially larger than the cladding index, n clad ∼ 1.46. Because of this core-confined mode profile, the mode area can be well approximated by the dimensions of the core itself, i.e. EMA ∼ = w × h, where h is the waveguide thickness and w is the waveguide width; thus explaining the linear mode expansion relationship. As a result of this linear behaviour, tapers used for a direct expansion almost exclusively use a linear tapering shape, other shapes have been considered showing only marginal improvements. [14] Furthermore, from the evaluated mode profiles ( Fig. 1 and 1(d); FULL) it can be seen that the amount of power located in the corner regions of the waveguide is negligible. This allows the use of the effective medium theory (EMT), which separates the field of the guided mode into Cartesian coordinates: ψ = ψ x ψ y . Using the EMT simplifies the structure to two simple one-dimensional slabs and reduces three-dimensional finitedifference time-domain (FDTD) simulations to a less computational intensive 2.5D FDTD. In Fig. 1(a), the EMT approximation (broken line) is applied in the direct expansion regime, with the calculated effective indices and mode profiles well approximated by the effective medium theory ( Fig. 1(c) and 1(d); FULL vs EMT), thus confirming the validity of the approximation in this region. From point (c) of Fig. 1(a) the mode area can also be increased through an indirect expansion by reducing the width of the waveguide. As w is reduced and reaches subwavelength dimensions, < λ /2n, the guided mode can no longer be tightly localized. As a result, the effective index of the waveguide abruptly drops, approaching that of the cladding material, leading to a strong, non-linear expansion of the mode area as seen in the (c)→(b) region. This mode expansion mechanism is often used to design nanophotonic mode converters for fiberchip coupling. [7,8,10,11] However, the EMT can no longer be applied in this subwavelength regime since the amount of power in the waveguide corner regions exceeds 80%, as seen at w = 150 nm in Fig. 1(b); FULL vs EMT. Although the EMT has been used in the inverse expansion regime [11,15], its use seriously compromises the results. Here, using a full vectorial solver, an inverse taper is designed for maximum coupling efficiency between a 4.9 µm mode-field diameter (MFD) fiber and a silicon waveguide (450 nm × 220 nm) for TE polarized light at λ = 1550 nm.

Taper tip optimization
The coupling efficiency between an optical fiber and a SOI waveguide is determined by two main factors: the overlap integral of the electromagnetic fields between the fiber and the SOI waveguide at the tip of the taper, and the efficiency of the mode conversion process between the taper tip and the interconnecting waveguide. Figure 2. shows the overlap integral between an optical fiber (MFD=4.9 µm; E i , H i ) and a silicon waveguide of different widths (E j , H j ). The overlap integral, η, is defined as: [16]  The overlap integral is maximized for w 0 of 155 nm and 100 nm at η = 0.92 (0.36 dB) for the quasi-TE and -TM modes, respectively. The overlap integral sets the upper limit of coupling efficiency since additional loss is expected as the mode propagates through the taper. In the calculation of the overlap integral, the magnetic field should not be simplified as a direct function of its electric counterpart because the SOI waveguide modes are not well approximated by propagating plane waves; they are quasi-TE/TM modes. [17] Accounting for the magnetic components of the guided modes leads to the calculation of optimal tip widths that are 20-30 nm larger than those previously reported [11].

Constant-loss framework
Now that the overlap integral is maximized at the taper's tip, the tapering function needs to be optimized for maximum light coupling efficiency between the tip and the interconnecting waveguide. The range of different, monotonically increasing taper functions to connect the taper tip (width, w 0 ) to the waveguide (width, w out ) in a given taper length, L, is bounded by the two  taper functions shown in Fig. 3(a) and 3(c). These piecewise functions are equivalent to each other, under the assumption that untapered waveguides are lossless. Given that these functions are abrupt, they represent the least efficient taper functions possible. In-between these cases an optimal taper function, that maximizes the efficiency can be inferred. Among the possible taper functions, the linear taper ( Fig. 3(b)) or the parabolic functions are often used [10,11], because of their simplicity but not necessarily optimal efficiency. To determine a taper shape with the highest possible efficiency, let us first define the total loss, L{w}, incurred by a taper function, w(z) and the instantaneous taper loss at point z as α(w,ẇ, z). The total loss of the taper can be expressed by the following functional: The optimal taper function w(z) that minimizes L{w} can be derived by solving the Euler-Lagrange differential equation: [18] Assuming that α does not depend explicitly on z, which is a reasonable hypothesis under the assumption that an untapered waveguide is lossless and that w(z) is a monotonically varying function, the following simplified equation is obtained: This implicit equation provides a clear guideline for designing an optimal taper function: the instantaneous loss at any point z within w(z) should be constant. Equation 5 establishes a general design principle that can be applied to engineer different types of photonic devices with varying shape functions. For an inverse taper, α(w,ẇ) depends on the coupling between the guided mode and the radiation modes. Given the infinite number of radiation modes, this calculation is computationally intensive. However, because at any point z along the taper only a single guided mode is supported, the instantaneous transmitted (T ) and reflected (R) powers between the guided modes at z− and z+ can be tracked. This approach is complementary to computing the instantaneous power loss to the radiation modes, i.e. α = 1 − R − T. Figure 3(d) schematically shows our taper design approach where the taper function is discretized into steps. The width of the first step, w 0 , is selected to maximize the overlap integral with the optical fiber mode (Fig. 2.); then, recursively, the taper is constructed by adding steps widths, δ i−1,i = |w i−1 − w i |, selected such that the loss penalty is constant, α i−1,i = α threshold . To compute the step loss, a coupled-mode theory is employed, leading to the following equations: [19,20] In practice, the reflection coefficient is several orders of magnitude smaller than the transmission coefficient and can often be neglected. In Fig. 4(b), at a given position z, the total power (P z ) is compared to the gaussian, 4.9 µm MFD source power, Γ z,source = P z /P source . Subsequently, the overlap integral between the field distribution and the local guided mode at z is calculated to obtain the power in the guided mode, P mode (z) = Γ z,source · η z,mode . In Fig. 4(b), all three tapers start with 92% of the source power guided in the tip mode (Fig. 2). In the linear taper, light is substantially dissipated near the tip as the mode propagates along the first 4 µm of the taper. The parabolic taper, whose width increases more slowly initially, yields an improved taper function. The best taper function is the constant-loss taper in which the mode conversion loss is more evenly distributed along the taper, in spite of the fast tapering rate in the 13-15 µm range ( Fig. 4(a)). The not quite linear power decay characteristic of the CLT in Fig. 4(b) is likely because a simple coupled-mode theory formalism was used to implement the constant-loss formalism; use of a finer 3D FDTD mesh could also improve the linearity of the loss profile. Halfway through the taper, the electric fields along the constant-loss taper and linear taper are compared in Fig. 4(d) and 4(e), respectively. In this region, the constant-loss taper is in an intermediate state of field delocalization while the mode conversion process of a linear taper is mostly completed, indicating that the constant-loss taper shape more efficiently uses the conversion length L available. Figure 5 shows the coupler efficiency as a function of taper length simulated in 3D FDTD. One can observe that the efficiency of the couplers increases with increasing taper lengths; furthermore, the CLT is found to be more efficient than either the parabolic or linear taper shape, for any taper length. It is worth nothing that the simulations that were used to compute the curves in Fig. 5 did not include a finite buried oxide layer (BOX) and therefore radiation losses to the substrate are neglected; i.e. Fig. 5 shows how the coupler efficiency is impacted by the mode transformation loss only. In actuality, radiation losses to the substrate would limit the maximal  achievable efficiency; as the tapers are made longer, the gain in efficiency that a smoother mode transformation process brings diminishes, as seen in Fig. 5 at longer taper length, while the radiative losses would steadily increase with length. For example, a 15-µm-long constant-loss taper has approximately the same efficiency mode transformation loss than a 22-µm; but the parabolic taper, being longer, would suffer from greater radiation losses. An important goal in designing practical edge coupling structures is to expand the mode size to match fibers with large mode field diameters in order to relax the fiber alignment tolerances. For typical BOX layer thicknesses used (2-3 µm) loss by light leaking into the substrate becomes a limiting factor to the overall coupling efficiency. These substrate losses can be minimized for a given mode size by minimizing the taper length, the constant-loss taper quantifies the miniaturization that can be achieved by optimal design of the widely used inverse taper couplers.

Taper robustness to fabrication
In Figure 5, the linear (and to a lesser extent the parabolic) taper exhibits a pronounced pointto-point jaggedness (four points are specifically shown). This is attributed to the 3D FDTD grid approximating the taper. Because the CLT's shape is close to minimizing the coupler's loss at each point, shape distortions incurred by the mesh affect less its efficiency compared to the linear shape. The mesh distortions effect is roughly analogous to the distortions that can be introduce in fabrication; therefore the CLT is expected to be more robust to fabrication.
In order to further demonstrate the robustness of the constant-loss taper to fabrication, 15- µm-long tapers were perturbed using an exponential autocorrelation roughness model [21] with a standard deviation of 25 nm and a autocorrelation length of 250 nm. A total of 100 perturbation profiles were generated for each coupler and then simulated in 3D FDTD. Figure 6 reports the normalized probability distribution of the coupler efficiency given our perturbation model. It can clearly be seen that not only is the constant-loss taper more efficient, but its distribution is more tightly grouped near the unperturbed taper's efficiency showing that the constant-loss taper is indeed more robust to fabrication than the linear or parabolic liner taper. The linear taper actually has some perturbed tapers (5 counts) that outperformed the unperturbed case; the efficiency of the linear taper is very sensitive to perturbation that occurs near its tip, as seen in Fig. 4(b), as a result perturbations that resulted in making the tip of the linear taper less abrupt resulted in a slightly better coupler. The median coupler efficiency are 5.0, 4.6, and 3.5 dB; respectively for a linear, parabolic, and constant-loss taper. In an industrial setting, producing components that have increased robustness to fabrication is key in improving one's fabrication yield, use of a constant-loss taper would result in improved taper fabrication robustness.

Experimental results
Inverted tapers were fabricated using a 220 nm silicon-on-insulator wafer with 3 µm buried oxide layer. The structures were defined using a hydrogen silsesquioxane (HSQ) resist with a inductively coupled, SF 6 − C 4 F 8 plasma etching and covered in a 3 µm thick plasma enhanced chemical vapor deposition (PECVD) oxide. In Fig. 5, at a ∼ 15 µm taper length, the expected efficiency of a constant-loss taper is noticeably higher compared to a linear or parabolic taper, therefore 15 µm long tapers were fabricated. Using a tunable laser and a polarization controller, light was coupled to a polarization-maintaining, 5 µm MFD or 3 µm MFD, lens fiber. The insertion loss of a single coupler was determined by measuring the total insertion loss to the chip, the total loss is composed of the loss from the input coupler, the waveguide, and the output coupler. The loss from the waveguide was de-embed from the the total insertion by measuring the waveguide loss through the cut-back method. Specifically, the waveguide loss was measure at 1.1 dB/cm or 0.53 dB for our 4800 µm long waveguide. This low waveguide loss enables precise measurement of the coupler's loss given that only 0.53 dB had to be deducted from the total insertion loss measurement in order to receive the couplers' loss. The measured efficiencies are reported in Fig. 5 for the 5 µm MFD fiber. The CLT is the best taper at 1.95 dB followed the parabolic, 2.21 dB, and linear, 3.25 dB. Measured efficiencies are slightly higher than simulated and this is attributed to the lens fiber used that collimates the beam to a 5 µm MFD spot. Using a 3 µm MFD fiber, the measured efficiencies improved respectively to 0.56 dB, 0.84 dB, and 0.95 dB. Use of the smaller MFD fiber resulted in better efficiency since radiation loss to the substrate are not as dominant as they are with a larger MFD fiber. Specifically, the efficiency of the constant-loss taper measured at 0.56 dB is remarkably high given its short 15 µm length.

Conclusion
A new design strategy of engineering losses constant through the taper has been introduced using a Lagrangian approach, and an original, constant-loss taper was derived. We showed with 3D-FDTD calculations that the constant-loss taper is both more efficient and robust than the linear and parabolic tapers. Finally, using a short 15-µm-long taper the efficiency of the constant-loss taper was measured at 0.56 dB, the highest efficiency per length ratio ever reported for a nanophotonic taper.