Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/D0NA00346H
(Paper)
Nanoscale Adv., 2020, Advance Article

Alina Karabchevsky*,
Eran Falek,
Yakov Greenberg,
Michael Elman,
Yaakov Keren and
Ioseph Gurwich

School of Electrical and Computer Engineering, Ben-Gurion University, Beer-Sheva 8410501, Israel. E-mail: alinak@bgu.ac.il

Received
1st May 2020
, Accepted 18th May 2020

First published on 3rd June 2020

Building blocks of photonic integrated circuitry (PIC), optical waveguides, have long been considered transparent. However, the inevitable Fresnel reflection from waveguide facets limits their transparency. This limitation becomes more severe in high-index waveguides in which the transparency may drop to 65%. We overcome this inherent optical property of high-index waveguides by engineering an appropriate facet landscape made of sub-wavelength artificial features unit cells. For this, we develop a semi-analytical formalism for predicting the metasurface parameters made of high-index dielectric materials, to be engraved on the facets of optical waveguides, based on Babinet's principle: either extruded from the waveguide facet or etched into it. Our semi-analytical model predicts the shape of anti-reflective metasurface unit cells to achieve transmission as high as 98.5% in near-infrared from 1 μm to 2 μm. This new class of metasurfaces may be used for the improvement of PIC devices for communication and sensing, where device transparency is crucial for high signal-to-noise ratios.

Despite the vast research published on optical transparency, very little is known about how to make optical waveguide transparent via sculpturing their facets with metasurfaces. For example, inside the absorption region one can use electromagnetic-induced-transparency (EIT) equivalent structures, made either of metals or all-dielectric materials e.g., silicon.^{12–15} These structures are utilizing the outside-generated energy to introduce a transparent region inside the absorption region. In a Si or GaAs waveguide, however, the excitation in a transparent region, and the losses they experience, are due to the reflection but not absorption.

One typical prior art solution to the efficiency problem mentioned above suggests the application of coatings such as aluminum, aluminum oxide, barium fluoride, calcium fluoride, cerium(IV) oxide, and others to the input and output facets of an optical waveguide. This typically reduces reflections in a specific spectrum range.^{16–21} However, the above coatings are not applicable to avoid reflections on the tiny surfaces of polished waveguide facets. Even where the deposition of a coating is possible, this solution still suffers from another problem: in harsh conditions, the coating tends to crack and even gets off the core. This is particularly noticeable in environments of high temperature, high-pressure or environmental stress changes, such as in airplanes, satellites, high-power devices and space stations. Matching oils are widely used to minimize Fresnel reflection. However, they are mainly suitable for a laboratory environments.

Changing the landscape of the waveguide facet at the nanoscale was proposed to avoid reflection;^{22–24} however, a broadband anti-reflection property for waveguides was never achieved. It is, therefore, the object of the present work to report on improving the efficiency of on-chip waveguides having high-index core, by obtaining broadband transparency. While developing the methodology to improve the transmission efficiency of waveguides, we set a goal: to study silicon particle-based metasurfaces^{25–31} in order to maximize the mean value of transmission at a given spectral range. The on-facet metasurfaces possess two fundamental properties: (1) a gradient index profile between two optically coupled media, and (2) a light trapping effect. To demonstrate the transparency effect in waveguides we developed the semi-analytical method, which allows for predicting the shape of the unit cell and its parameters.

The following sections provide the theory as well as numerical proof-of-concept simulations exploring the properties of dielectric metasurfaces engraved in – or extruded from – the waveguide facets, to study the waveguide transparency effect.

It is well known that for a single wavelength, one can obtain almost zero reflection.^{33} However, any widening of the spectral range affects the reflection at this specific wavelength. In the present analytical description, we consider the minimization of the mean value reflection at a given spectral range, together with minimizing the deviation value.

The choice of the unit cell for the metasurface is also not trivial since neither the shape nor the geometry are prescribed by general theoretical arguments. Therefore, let us consider a periodic arbitrarily shaped metasurface on a waveguide facet (Fig. 1a).

All parameters of the considered structure, except its total width, are small in comparison to the incident wavelength λ in a vacuum.

Next, we divide the facet into infinitesimally thin layers y_{n}. Since the structure is sub-wavelength, the formalism of effective medium theory can be used. In addition, under the geometrical optics approach, we approximate the description to one in which the higher-order scattering is ignored. Assuming the period of the structure is X < λ, the height is designated as H (which is unknown at this point), and the refractive index of the medium as m_{h}. If we denote σ_{n} as the part of the period filled by the medium in the n^{th} layer, then the effective refractive index in this layer can be estimated as:

(1) |

Considering the normal incidence illumination, the reflection r_{n} from the boundary between n^{th} and (n + 1)^{th} layers is

(2) |

At the upper surface of the facet (blue solid line in Fig. 1), the contribution of r_{n} is

(3) |

The wave number k(y) can be written as

(4) |

Generally, the reflection of a layered structure is represented by the bulk relation, where the analysis of such representation is a relatively cumbersome task. Here, we restrict ourselves by assuming that: |r_{n}| ≪ 1, and also that the transmitted wave approximately preserves its amplitude; in other words, |t_{n}| ≈ 1. In the framework of this approximation, the total reflection can be written as , and represented by the integral as

(5) |

The absolute value of R_{T} has to be minimized. We solve this minimization problem for a monochromatic wave. Dealing with a fixed H, the problem can be defined as a variational one. The shape of the unit-cell, which is represented by σ(y), will now be determined. Suppose δσ is a small variation of σ(y); then the total reflection for this new unit-cell is

(6) |

(7) |

(8) |

For simplicity, we denote the function σ(y) by σ and γ(y) by γ. In terms of γ(y), eqn (6) takes the form of

(9) |

For obtaining γ(y) in a closed form, we assume that the function is an order smaller than ‖δγ‖ (o(‖δγ‖)), where ‖δγ‖ is a certain norm of δγ in the interval y ∈ [0, H]. While this may be not correct for an arbitrary δγ, and an arbitrary y, at least for small H, the integral contribution of wrong y points will be small.

In the framework of the above-mentioned assumption , we obtain the variation of R_{Tδ} as

(10) |

After integrating the term by parts, the real part vanishes, and we obtain a simplified form of δR_{Tδ} as

(11) |

According to calculus of variations, we set δR_{Tδ} = 0, and thus, the eqn (11) leads to

(12) |

In eqn (12), the point y = 0 is singular and should be treated with care, since also γ − 1 = 0, and thus, (γ − 1)/y are undefined at y = 0. Therefore, one should choose a trajectory for integration in the complex plane. By doing so, the solution γ = 1 + Cy is obtained, and the integration constant C can be taken as 1 without lost of generality.

Then, for σ(y) the following is obtained:

(13) |

Eqn (13) describes a linear dependence of the y-level fill-factor on y. It shows that each unit-cell has a constant slope. This prescribes a conical or pyramidal shape of the unit-cells as shown in Fig. 1b.

Analytically obtained optimal shape of a unit-cell providing anti-reflection can be found in the state of the art experimental studies.^{22,35–38} We stress, however, that although explored experimentally, the correct parameters of these structures have not yet been obtained as optimal ones to suppress reflection in a broad spectral range.

A linear function for the filling fraction σ(y) implies that the effective refractive index is also linear in y according to eqn (1). In Fig. 2 we show the transmittance as a result of a plane-wave incident on a half plane with refractive index n_{s} = 3.5. At the interface of the incidence we added a gradient index matching layer with thickness H. The matching layer was divided into slices with a thickness of 10 nm, where every slice had a refractive index according to the linear function

(14) |

The structure was simulated based on transfer-matrix method^{39} to obtain the transmittance of this structure. In Fig. 2 one can notice that the transmittance is significantly higher than 69%, which is the direct transmittance from a half-plane with n = 3.5. One can also notice that the transmittance does not exhibit any resonant behavior. In other words, the anti-reflection property holds for wide bandwidth.

By taking into account higher-order effects such as multiple scattering, the shape can be modified, and the new form can be calculated by solving numerically the wave-equation for waveguide systems. However, if we consider that σ(H) = X, it is immediately obtained from eqn (13) that H = X(m_{h} − 1). Therefore, the above assumption with respect to a small value of H is justified for m_{h} close enough for 1. This justifies also the main assumption about the small value of |r_{n}|, and thus the possibility of ignoring the multiple scattering.

Based on the gradient index (GRIN) optics, it can also be stated that the lowest reflection can be achieved by structuring the entire area of the waveguide facet with the metasurface. This means that the optimal filling factor O_{n} of the facet prescribes unit-cells being densely distributed on the facet. Otherwise, a step change of the refractive index would occur at the layer where y = H. The arguments above also show a weak effect of the randomization of the unit-cells location on the facet: such a process generally contradicts the requirement of dense spatially distributed unit-cells within the metasurface.

The trapping of photons also contributes to the transmission. Therefore, the value of H predicted above turns out to be overestimated. It is also expected that multiple scattering would contribute more in the bottom layers than in the upper ones. It is plausible to estimate the likelihood of every trapped photon (and thus, contribution to transmission) by the additional factor of ρ^{2}τ/s, where ρ is the reflectance of each unit-cell, τ is the transmittance through the metasurface, and s is the distance between interacting unit-cells. This difference between the upper base d_{u} and the bottom base d_{b} in the multi-scattering process may be corrected by fitting the shape of the truncated cone by a paraboloid.

Based on our theoretical evaluations described in the previous sections, we found that a periodic lattice made of densely distributed cones minimizes the reflection effect. To prove this, we built a full-wave numerical simulation using Lumerical and CST Finite-Difference-Time-Domain (FDTD) solvers and calculated the optical characteristics of an arbitrary waveguide. In our numerical investigation, we consider the rib waveguide with the structure depicted in Fig. 3a. This contains the silicon guiding layer (Si, n = 3.4784 (ref. 40)) on silicon-dioxide substrate (SiO_{2}, n = 1.444 (ref. 41)) with rib width of W = 10 μm, height T = 2.4 μm and rib height t = 0.5 μm. We note that the unit cell is extruded from the same material as the waveguide. The calculated fundamental TE mode profile is shown in Fig. 3e.

It is important to stress, that even though the silicon waveguides can be produced with smaller cross-sections, which is commonly used to support a single mode propagation, the proposed AR structure is not mode dependent and works well also for the multi-mode waveguides. In addition, as the cross-section area increases, the array of cones becomes larger, which improves the collective effect of the structure as a metasurface, and reduces the effects caused by the edges. Smaller cross-section would require smaller unit-cells, which is more challenging to fabricate. For these reasons a larger cross-section was chosen in our simulation. In the numerical model, we launch the Gaussian beam into the waveguide with a radius of 1.5 μm where the waist (focal point) is 1.5 μm from the source. The beam is directed in parallel to the waveguide axis z and polarized linearly to the y direction. The facet is placed at a distance of 1.5 μm where the beam focuses. The center of the beam is aligned with the center of the waveguide guiding layer such that maximal power is incident upon the facet. This anticipates realistic experimental conditions, where fiber is butt-coupled to the waveguide. Since the beam spot profile is larger than the dimensions of the waveguide facet, it leads to the beam-facet mismatch losses. We calculate the transmission by integrating the power flow in the z direction at a distance of 2 μm from the facet (the plane of the cone bases). The integration is done over the area of the facet, i.e., x ∈ [−5, 5] μm, and y ∈ [0, 2.4] μm. The integrated power was normalized to 1 watt, which is the total power of the Gaussian beam source.

To estimate the sensitivity to different structural parameters of the conical shape unit cells, we modeled a hexagonal lattice of truncated cones (Fig. 3b) placed on the facet and adjacent to each other such as in Fig. 3a. The hexagonal grid was chosen over a rectangular one since it is more efficient due to the larger fill-factor and was chosen for our numerical test as seen in Fig. 3a. The transmission spectrum as a function of wavelength for several cone heights H is shown in Fig. 4a. The cone base d_{b} and upper d_{u} diameters are kept constant at d_{b} = 550 nm and d_{u} = 200 nm, respectively.

One can see from Fig. 4a that as the cone height increases, the overall average transmission increases as well. Although theoretically higher cones may produce better performance, practically, the larger aspect ratio of the cones (height vs. base diameter) is limited due to fabrication constraints. One can also notice the dip at 1100 nm, which is a result of a resonance due to the periodic structure. At that particular wavelength, the period of the structure is exactly λ/2. To understand the influence of height on the transmission, while neglecting the collective effect, we calculated the scattering angular diagrams of the unit cells of truncated conical shape (Fig. 3b) for the wavelength of 1100 nm from Fig. 4a. Inset of Fig. 4a shows that when the particle height increases, backward scattering is suppressed while allowing the angular dependence of the forward scattering effect.

Next, we study the sensitivity of the transmission to the base diameter d_{b} of the unit-cell. Fig. 4b shows the transmission spectrum as a function of wavelength for an hexagonal array of truncated cones with varying base diameters while the height and upper diameters are fixed at H = 900 nm and d_{u} = 200 nm, respectively.

Base diameter of d_{b} = 550 nm (red line in Fig. 4b) exhibits the highest transmission over the chosen spectral range with = 96.2% with H = 900 nm. The base diameter is closely related to the periodicity of the structure, which in turn affects the spectral behavior of the device. The change of the base diameter affects the filling factor and therefore affects the transmission.

The transmission of = 96.2% was obtained for the cone height of H = 900 nm and the base diameter of d_{b} = 550 nm.

Decomposing a unit cell of the periodic grid made of truncated cones into its Fourier components can explain why it yields worse performance characteristics compared to the cones with the hemispherical tip or of paraboloidal shape. The decrease in performance occurs since more energy is carried in higher diffraction orders. The orders destructively interfere with each other and reduce the overall transmission. Thus, the smoothing of the sharp edges of the cones results in a more uniform transmission in a required spectral band.

The transmission spectrum as a function of wavelength for different cone shapes is shown in Fig. 5a with the cones distributed in a hexagonal grid. This allows comparison of the performance characteristics for different tip shapes, such as truncated tip and a parabolic tip, as shown in Fig. 3b–d. In addition, we maximized the filling factor of the facet by decreasing the base unit-cell diameter d_{b}, against the previous estimations. As a result, the diameter leading to the highest transmission occurred when d_{b} = 450 nm. The height and upper diameter for the truncated tip cones are kept constant at H = 900 nm and d_{u} = 200 nm, respectively.

Fig. 5a shows the averaged transmission as high as 98.13% in the wavelengths range from 1 μm to 2 μm using cones with parabolic tips (Fig. 3d) in the hexagonal grid. Fig. 5b shows the reflection spectrum from the input waveguide facet. The facet has an hexagonal grid of conical unit cells with d_{b} = 450 nm and H = 900 nm.

While changing the shape of the unit-cells, we analyze the results based on the diffraction grating model. Minimizing the overall reflection involves minimizing the reflection of some orders of this grating. This can also be formulated as a π phase shift at the diffraction angle β = π/2. Since the beam is incident upon the facet from the air medium, the requirement dictates the relation H = λ_{0}/2 for a chosen wavelength of λ_{0}. Considering the spectral range λ ∈ [λ_{min}, λ_{max}], it would be natural to take λ_{0} = (λ_{min} + λ_{max})/2. However, for a wide spectral range, this can lead to an appearance of the reflection maximum at the shortest wavelength.

We note that the structures we predicted can be fabricated using conventional focus ion beam (FIB) machine. Beginning with a smooth facet, the FIB machine can etch the facet based on a pre-programmed computer-aided design (CAD), and remove material, to produce the facet as in Fig. 3. For example, a FIB machine, Thermo Fisher Scientific, dual-beam G3, can be used to create the truncated cones.

First, we explore the influence of truncated cone shapes on AR properties of the waveguide facet. To this end, we integrate the power flow in the z direction behind the structure, at a distance of 2.5 μm from the facet. The choice of this distance is dictated by the overall simulation sizes. The unit-cells distribution is chosen to be hexagonal, the same as was done in the extruded structures case. In addition, we integrate the power inside the waveguide (the transmitted power) for 4 planes at a distance of 9–13 μm from the facet. Then we average the transmitted power to eliminate noise. We note that for shorter wavelengths, the unit-cell size is not sub-wavelength in terms of the wavelength propagating in silicon. Therefore, at these wavelengths, the structure experiences near-field effects due to diffraction. We noticed that for the short wavelengths the power inside the waveguide is unstable for a distance of approximately 6 μm from the facet. For this reason, we calculate the transmitted power as far as 9 μm from the facet. The waveguide structure was terminated after 14 μm with an absorbing boundary condition (perfectly matched layer-PML) so that the simulated structure behaves as an infinite waveguide.

Fig. 8a shows a comparison of the transmittance as a function of wavelength. The diameter of the truncated cones is 450 nm and the depth is varied from H = 800 nm to H = 950 nm in steps of 50 nm. As expected from Babinet's principle, the calculated results are very close to those shown in Fig. 4a, obtained for the extruded features. The transmission drops for longer wavelengths since H becomes shorter in terms of wavelength at these regions. The decrease in transmission can also be seen in Fig. 2, where we simulated a plane-wave incident on an infinite gradient index layer. The lower transmittance can be attributed to several factors, such as unit cell size and discontinuity in the refractive index. The size of the unit-cells is not small enough for the structure to operate in shorter wavelengths due to the large refractive index of the waveguide core material (silicon in our case). Also, the effective refractive index is not continuous at the air-metasurface interface and at the metasurface-waveguide core interface, since the filling factor in cone structures cannot reach 100% with round features.

Next, we compare the effect of the unit-cell size in Fig. 8b. Here, we consider the depth of the cones as a constant value of 900 nm and vary the base diameters for values of d_{b} = 350, 450, 550, and 650 nm. As expected, the shorter wavelengths region shows higher transmission as the structure approaches the sub-wavelength regime. However, the results for d_{b} = 350 nm also exhibit stronger resonant behavior due to the increase in the effective index at the interfaces. Even though a unit-cell size of d_{b} = 350 nm can increase transmission by up to 99.4%, one can achieve transmission above 95% over a spectral range of 1–1.8 μm by increasing the unit cell by up to d_{b} = 450 nm.

Now, we compare the truncated cone holes with a diameter of d_{b} = 450 nm and height of H = 900 nm, to paraboloids with the same depth and base diameter. Fig. 8c shows the result of this comparison. We note that the AR structure on a facet made of paraboloids exhibits less resonant behavior, due to the continuity of the effective index at the tip of the metasurface. By using a parabolic array of holes one can achieve transmission of more than 94% over the entire spectral range of 1–2 μm.

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