BY 4.0 license Open Access Published by De Gruyter August 13, 2021

A review of silicon subwavelength gratings: building break-through devices with anisotropic metamaterials

José Manuel Luque-González ORCID logo, Alejandro Sánchez-Postigo, Abdelfettah Hadij-ElHouati, Alejandro Ortega-Moñux, J. Gonzalo Wangüemert-Pérez, Jens H. Schmid, Pavel Cheben, Íñigo Molina-Fernández and Robert Halir
From the journal Nanophotonics

Abstract

Silicon photonics is playing a key role in areas as diverse as high-speed optical communications, neural networks, supercomputing, quantum photonics, and sensing, which demand the development of highly efficient and compact light-processing devices. The lithographic segmentation of silicon waveguides at the subwavelength scale enables the synthesis of artificial materials that significantly expand the design space in silicon photonics. The optical properties of these metamaterials can be controlled by a judicious design of the subwavelength grating geometry, enhancing the performance of nanostructured devices without jeopardizing ease of fabrication and dense integration. Recently, the anisotropic nature of subwavelength gratings has begun to be exploited, yielding unprecedented capabilities and performance such as ultrabroadband behavior, engineered modal confinement, and sophisticated polarization management. Here we provide a comprehensive review of the field of subwavelength metamaterials and their applications in silicon photonics. We first provide an in-depth analysis of how the subwavelength geometry synthesizes the metamaterial and give insight into how properties like refractive index or anisotropy can be tailored. The latest applications are then reviewed in detail, with a clear focus on how subwavelength structures improve device performance. Finally, we illustrate the design of two ground-breaking devices in more detail and discuss the prospects of subwavelength gratings as a tool for the advancement of silicon photonics.

1 Introduction

Optical metamaterials are synthetic structures with physical properties that are not readily accessible in nature. The most exotic metamaterial properties are attained via resonant structures, which induce substantial changes in the properties of light over distances comparable to the free-space wavelength, thereby enabling the realization of ultra-thin metasurface components [1], [2], [3], [4], [5], [6]. Nonresonsant dielectric structures at the subwavelength scale, on the other hand, behave as effective homogeneous media with an anisotropic refractive index tensor [7]. Subwavelength grating metamaterials, since their first demonstration in integrated silicon waveguides at NRC Canada [8], [9], [10], [11], [12], [13], [14], [15], [16], are playing a crucial role in the field of silicon photonics [17, 18]. Silicon photonics leverages the advanced nano-fabrication infrastructure established by the complementary metal–oxide–semiconductor (CMOS) microelectronics industry to drive key advances in areas as diverse as datacom and telecom [19], metrology [20], biochemical sensing [21, 22], quantum signal processing [23], and light detection and ranging (LiDAR) [24]. While the large refractive index contrast between silicon (n Si ≈ 3.5) and its native oxide ( n SiO 2 1.45 ) enables dense integration, the scarcity of CMOS-compatible materials with intermediate refractive indices restricts the design space. This limitation can be overcome by structuring silicon waveguides at the subwavelength scale. Using a single, full etch step, a wide range of equivalent anisotropic materials can be synthesized [18]. This concept is schematically illustrated in Figure 1, which compares the propagation through a silicon waveguide structured at subwavelength scale, and the propagation through the equivalent anisotropic waveguide. Indeed, the flexibility enabled by silicon subwavelength grating (SWG) structures has led to breakthrough advances in many areas of silicon photonics, including broadband, ultra efficient fiber-to-chip couplers [25], [26], [27], high-performance integrated filters [28, 29], power and polarization splitters covering several optical communication bands [30, 31], on-chip graded-index (GRIN) lenses [32], evanescent field control [33], or even the exploration of non-Hermitian systems [34]. At telecom wavelengths, the feature sizes required for subwavelength operation are ∼100 nm, which is why most subwavelength-based devices reported so far have been fabricated with e-beam lithography. However, significant advances in deep-ultraviolet immersion lithography now enable volume fabrication of structures with sub-100-nm feature sizes [35, 36], and several subwavelength-based devices are already mass fabricated [37], [38], [39].

Figure 1: 
Simulated propagation of the electric field through (a) a tilted subwavelength waveguide and (b) the equivalent homogeneous anisotropic waveguide. Linear polarization in the x–z plane and a free-space wavelength of 1.55 μm are considered. In both (a) and (b), the geometry of the waveguide is outlined in black. The waveguide layer is 220 nm thick and 3 μm wide. In (a), Λ = 250 nm, DC = 0.5 and θ = 30°. In (b) the permittivity tensor of the waveguide core metamaterial is shown.

Figure 1:

Simulated propagation of the electric field through (a) a tilted subwavelength waveguide and (b) the equivalent homogeneous anisotropic waveguide. Linear polarization in the xz plane and a free-space wavelength of 1.55 μm are considered. In both (a) and (b), the geometry of the waveguide is outlined in black. The waveguide layer is 220 nm thick and 3 μm wide. In (a), Λ = 250 nm, DC = 0.5 and θ = 30°. In (b) the permittivity tensor of the waveguide core metamaterial is shown.

As the pace of progress in the field of silicon subwavelength structures continues to accelerate, in this review we aim to provide a comprehensive and insightful overview of the latest developments. Following the approach of our previous reviews on the topic [18, 40, 41], we start with an in-depth discussion of the most up-to-date knowledge of the foundations of subwavelength structures: in Section 2 we study to how and to what extent these structures can be modeled as homogeneous anisotropic metamaterials. The optical properties of these metamaterials are instrumental in understanding how subwavelength structures are exploited to enhance each family of devices we revise in Section 3: waveguides, broadband couplers, beam expanders, polarization controllers, filters, wavelength- and mode-division multiplexing devices, fiber-chip couplers, optical antennas, and evanescent field sensors. In Section 4 we simulate a small selection of cutting-edge subwavelength-based devices, emphasizing how the optical properties of subwavelength structures are exploited and illustrating how the anisotropic modeling and the appropriate electromagnetic tools help efficiently design those devices. Finally, in Section 5 conclusions are drawn and a short outlook is provided.

2 Understanding silicon subwavelength metamaterials

The core idea underlying subwavelength gratings is that they enable us to synthesize integrated optical materials with tailorable properties. To exploit the full potential of these structures, it is crucial to understand under which conditions periodic structures act as equivalent homogeneous materials and what exactly the properties of these materials are.

To illustrate the main goal of this section, let us examine the segmented silicon waveguide shown in Figure 2(a). This structure is periodic in the longitudinal direction (z) and is therefore described rigorously by modal analysis using the Floquet–Bloch theory. Light propagates through this dielectric waveguide with no loss when the period (Λ) is below the Bragg threshold, i.e., Λ < λ 0/(2n eff), where λ 0 is the operating wavelength (in vacuum) and n eff is the effective index of the fundamental Floquet–Bloch mode supported by the structure. Under this condition, the periodic waveguide operates in the subwavelength regime and effectively behaves as a solid-core waveguide with the equivalent core material that is described by the permittivity tensor ε = diag [ n x x 2 , n y y 2 , n z z 2 ] . The technique that yields the values of the tensor components is known as homogenization and is used to model the actual subwavelength waveguide as a homogeneous anisotropic metamaterial waveguide. The simulation of the equivalent, homogenized structure is substantially less time-consuming than rigorous Floquet–Bloch analysis of the SWG waveguide and provides useful physical insight on how SWG-based high-performance integrated devices work. In this section, we will study the behavior of subwavelength waveguides and how to accurately homogenize them.

Figure 2: 
(a) Schematic of a longitudinal SWG waveguide. The period of the structure, Λ, is much shorter than the guided wavelength, λ, of the light propagating along the z-axis. (b) Schematic of the homogeneous waveguide model of the structure shown in (a). The SWG waveguide core is modeled as a homogeneous anisotropic medium with a permittivity tensor 
ε
.

Figure 2:

(a) Schematic of a longitudinal SWG waveguide. The period of the structure, Λ, is much shorter than the guided wavelength, λ, of the light propagating along the z-axis. (b) Schematic of the homogeneous waveguide model of the structure shown in (a). The SWG waveguide core is modeled as a homogeneous anisotropic medium with a permittivity tensor ε .

The accuracy of the homogenization process decreases as the ratio Λ/λ approaches the Bragg regime since the resonances within the photonic bandgap cannot be described by a homogeneous metamaterial. In the following we will present two modeling methods and compare the results they provide with the rigorous Floquet–Bloch solution of the subwavelength waveguide:

  • An analytical laminar model of the subwavelength material, which is adequate for most applications as long as the vicinity of the Bragg threshold is avoided (Section 2.1).

  • A more sophisticated slab model that takes into account the thickness of the silicon waveguide layer and provides accurate results even when the waveguide operates near the Bragg regime (Section 2.2).

These models will be used to approximate subwavelength waveguides with longitudinal (z-periodic) and transverse (x-periodic) segmentations, while more complex (tilted) periodic patterns will be analyzed in Section 2.3.

Throughout this section we will assume typical material parameters for silicon photonics: n Si = 3.476, n SiO 2 = 1.444 , a duty cycle directional coupler (DC) = a/Λ = 0.5, and a silicon thickness H = 220 nm, operating at a wavelength λ 0 = 1.55 μm. The width of the waveguide is W = 3 μm. For illustration purposes we consider pitches of Λ = 50 nm, which is in the long-wave limit (Λ ≪ λ), and Λ = 220 nm, which can be readily fabricated with modern lithography techniques and is still below the Bragg threshold, Λ Bragg wg = 300  nm for our exemplary waveguide. For the sake of simplicity, we assume that the cladding and substrate are the same materials. We furthermore refer to light polarized along the y-axis as transverse magnetic (TM) polarized, and light polarized along the x- or z-axis as transverse electric (TE) polarized.

2.1 Laminar material model

The laminar model considers the subwavelength structure as a cut-out of an infinite laminar arrangement of two materials with period Λ, as illustrated in Figure 3(a). The material laminae have refractive indices n 1 = n Si and n 2 = n SiO 2 , and lengths a = DC·Λ and b = (1 − DC)·Λ, respectively. To obtain the equivalent permittivity tensor of the laminar material, we take into account the propagation of TE and TM plane waves in the xz plane. Specifically, we study the wave vector, k = k x x ˆ + k z z ˆ , for different propagating angles ϕ = tan−1(k x /k z ) [see Figure 3(a)]. The wave vector components (k x and k z ) are calculated using the dispersion relations [42]:

(1) cos ( k z Λ ) = cos ( k 1 z a ) cos ( k 2 z b ) Δ TE sin ( k 1 z a ) sin ( k 2 z b ) ,

(2) cos ( k z Λ ) = cos ( k 1 z a ) cos ( k 2 z b ) Δ TM sin ( k 1 z a ) sin ( k 2 z b ) ,

for TE and TM polarizations, respectively, where k i z = ( k 0 n i ) 2 k x 2 , and Δ is a polarization-dependent factor:

(3) Δ TE = 1 2 ( n 2 2 n 1 2 k 1 z k 2 z + n 1 2 n 2 2 k 2 z k 1 z ) , Δ TM = 1 2 ( k 1 z k 2 z + k 2 z k 1 z ) .

Figure 3: 
(a) Laminar periodic structure comprising two transversely infinite materials arranged along the z-axis with a period Λ. The laminae have lengths a = DC⋅Λ and b = (1 − DC)·Λ, and refractive indices n
1 and n
2. (b), (c) Comparison between the normalized wave vector components, k

x
 and k

z
, of a plane wave propagating along any angle of the laminar periodic structure (DC = 0.5) and of its equivalent anisotropic homogeneous material for the structure operating in the (b) long-wave regime, Λ = 50 nm, and (c) subwavelength regime, Λ = 220 nm. (d) Dependence with the duty cycle of the permittivity tensor components n
∥ and n
⊥ of the equivalent anisotropic homogeneous structure.

Figure 3:

(a) Laminar periodic structure comprising two transversely infinite materials arranged along the z-axis with a period Λ. The laminae have lengths a = DC⋅Λ and b = (1 − DC)·Λ, and refractive indices n 1 and n 2. (b), (c) Comparison between the normalized wave vector components, k x and k z , of a plane wave propagating along any angle of the laminar periodic structure (DC = 0.5) and of its equivalent anisotropic homogeneous material for the structure operating in the (b) long-wave regime, Λ = 50 nm, and (c) subwavelength regime, Λ = 220 nm. (d) Dependence with the duty cycle of the permittivity tensor components n and n of the equivalent anisotropic homogeneous structure.

These equations fully characterize the propagation of a plane wave along the laminar periodic structure for any period-to-wavelength ratio. However, in this review we focus on the subwavelength regime, which for the particular laminar structure we are considering implies using a pitch Λ < Λ Bragg lam = 240  nm . Since the laminar material is infinite, the effective indices of the Floquet–Bloch modes are larger than in the subwavelength-structured waveguide, and hence the Bragg threshold of the laminar material is smaller than that of the subwavelength waveguide ( Λ Bragg lam < Λ Bragg wg ). In any case, in the long-wave limit, Λ ≪ λ, we have k z Λ ≪ 1 (and hence k 1z a ≪ 1 and k 2z b ≪ 1), so that Eqs. (1) and (2) simplify to

(4) k x 2 n 2 + k z 2 n 2 = k 0 2 ,

(5) k x 2 n 2 + k z 2 n 2 = k 0 2 .

Equations (4) and (5) are the dispersion relations of TE and TM plane waves propagating along the xz plane of a uniaxial anisotropic crystal [43], with n and n ordinary and extraordinary indices of refraction respectively. Note that Eq. (4) corresponds to an ellipse while Eq. (5) is a circle. That is to say, a plane wave propagating along a laminar periodic structure in the long wave regime behaves as if it propagated along a homogeneous uniaxial crystal defined by the diagonal permittivity tensor ε l:

(6) ε l = [ n 2 0 0 0 n 2 0 0 0 n 2 ] .

The parameters n and n are the effective indices of a TE plane wave propagating along the z-axis and the x-axis of the laminar periodic structure, respectively, and can thus be defined from Eq. (1) as

(7) n = k z / k 0 ( when k x = 0 ) ,

(8) n = k x / k 0 ( when  k z = 0 ) .

In the Supplementary material we provide MATLAB and Python scripts to calculate the indices n and n . For the complete modeling of the metamaterial, we are only considering the equation for TE polarization [Eq. (1)], as it characterizes both required indices. Remarkably, the n parameter also yields a good description for TM polarization, because TM-polarized light is parallel to the laminae irrespective of the direction of propagation. In Figure 3(b) and (c) we compare the wave vector of a plane wave propagating along the laminar materials and along its equivalent anisotropic structure in the long-wave regime, Λ = 50 nm, and for a practical pitch, Λ = 220 nm, for DC = 0.5. In both cases, the pairs (k x , k z ) are obtained from Eqs. (1) and (2) for the laminar periodic structure and from Eqs. (4) and (5) for the equivalent anisotropic medium.

In the long-wave regime [Figure 3(b)], the laminar SWG and its homogenized model exhibit the exact same behavior. Moreover, it can be shown that in this regime the metamaterial properties do not depend on the period nor the wavelength, allowing us to simplify the permittivity tensor components as [7]:

(9) n 2 DC · n 1 2 + ( 1 DC ) · n 2 2 ,

(10) n 2 ( DC · n 1 2 + ( 1 DC ) · n 2 2 ) 1 .

As intuition predicts, a high duty cycle implies permittivity tensor components close to n 1 and a low duty cycle implies tensor components close to n 2 [see Figure 3(d)]. Indeed, this is the principle which enables refractive index engineering using subwavelength materials.

For the larger 220-nm pitch, the laminar periodic structure and the anisotropic metamaterial still yield a remarkably similar result, implying that the latter is a good model for the former even when the long-wave condition is not satisfied [see Figure 3(c)]. Even the duty-cycle dependence of n and n closely follows the results obtained in the long-wave regime [see Figure 3(d)].

Having discussed the properties of infinite laminar materials, we return to our main goal: understanding to what extent subwavelength-patterned waveguides, such as those in Figure 4(a) and (e), can be modeled by homogeneous anisotropic metamaterial waveguides. For this purpose, the modes of the subwavelength waveguide have been calculated rigorously using a 3D finite-difference time-domain (FDTD) software for photonic structures [44]. The parameters n and n of the metamaterial waveguide have been calculated using the script given in the supplementary material. The modes of the resulting anisotropic waveguide were then obtained using a simple finite element method (FEM) mode solver for photonic structures [45].

Figure 4: 
(a) Schematic representation of a longitudinal SWG waveguide. (b)–(d) Effective indices of (b) the fundamental Floquet–Bloch mode and (c), (d) higher order Floquet–Bloch modes of a longitudinal SWG waveguide calculated by rigorous 3D full-vectorial simulations (TE: Blue solid lines, TM: Orange solid lines) compared with those of the corresponding anisotropic homogeneous-core waveguides obtained using the laminar model (circles) and the slab model (triangles). (e) Schematic representation of a transverse SWG waveguide. (f)–(h) Effective indices of (f) the fundamental Floquet–Bloch mode and (g), (h) higher order Floquet–Bloch modes of a transverse SWG waveguide calculated by rigorous 3D full-vectorial simulations (TE: Blue solid lines, TM: Orange solid lines) compared with those of the corresponding anisotropic homogeneous-core waveguides obtained using the laminar model (circles) and the slab model (triangles).

Figure 4:

(a) Schematic representation of a longitudinal SWG waveguide. (b)–(d) Effective indices of (b) the fundamental Floquet–Bloch mode and (c), (d) higher order Floquet–Bloch modes of a longitudinal SWG waveguide calculated by rigorous 3D full-vectorial simulations (TE: Blue solid lines, TM: Orange solid lines) compared with those of the corresponding anisotropic homogeneous-core waveguides obtained using the laminar model (circles) and the slab model (triangles). (e) Schematic representation of a transverse SWG waveguide. (f)–(h) Effective indices of (f) the fundamental Floquet–Bloch mode and (g), (h) higher order Floquet–Bloch modes of a transverse SWG waveguide calculated by rigorous 3D full-vectorial simulations (TE: Blue solid lines, TM: Orange solid lines) compared with those of the corresponding anisotropic homogeneous-core waveguides obtained using the laminar model (circles) and the slab model (triangles).

Figure 4(b) compares the effective indices of the fundamental TE and TM modes of the longitudinal subwavelength waveguide [Figure 4(a)] with those of the metamaterial waveguide, as a function of the structural pitch. We observe that the laminar model yields a satisfactory estimation of the effective indices for both the fundamental TE and TM modes up to a pitch of ∼200 nm. Beyond that point, the infinite laminar structure approaches its Bragg threshold Λ Bragg lam = 240  nm , and can no longer represent the subwavelength waveguide, which has a significantly larger Bragg threshold Λ Bragg wg = 300  nm . The propagation constants of higher-order modes are compared in Figure 4(c) and (d). As expected, in the long-wave limit (Λ = 50 nm) the anisotropic model is in very good agreement with the exact Floquet–Bloch solution. Remarkably, even for a comparatively large pitch of 220 nm, the anisotropic model provides a very good description of higher-order modes for both polarizations with a maximum error in the effective index below 6%.

Figure 4(f) compares the effective indices of the fundamental TE and TM modes of the transverse subwavelength waveguide [see Figure 4(e)] with those of the metamaterial waveguide, as a function of the structural pitch. We observe a very good agreement in the long-wave regime, with the results of the metamaterial waveguide becoming slightly less accurate as the pitch is increased. This mismatch is especially remarkable for the TM polarization, as it mainly depends on the n component of the tensor, which is the component affected by the Bragg regime. For the higher-order modes, the propagation constants are very well predicted in the long-wave regime [see Figure 4(g)] and exhibit an error below ∼3% for a 220-nm pitch [see Figure 4(h)].

2.2 Slab material model

The main purpose of the slab material model is to overcome the limitations of the laminar model in the vicinity of the Bragg threshold. This is achieved by considering the silicon thickness H as a parameter of the material itself [see Figure 5(a)]. Following a similar argument as in the previous section, we calculate the permittivity tensor components from the propagation of two TE-polarized plane waves propagating along the x- and the z-axis. As opposed to the laminar model, for which analytical formulas are available, these calculations have to be performed numerically. Using the slab model is still advantageous over the rigorous 3D Floquet–Bloch analysis of the subwavelength-structured waveguide, because the calculations required for the model can be performed very efficiently using 2D plane-wave expansion method simulations [46]. As a result, we obtain the effective indices of the fundamental TE slab mode propagating along the z- and the x-axis, n eff x and n eff z [see Figure 5(b)]. Here the subindices indicate the polarization and not the direction of propagation. In order to convert these effective indices into refractive indices describing the metamaterial, we solve an isotropic slab for different core material refractive indices n eq, obtaining the mapping function shown in Figure 5(c). Finally, we map the effective indices n eff x and n eff z into the equivalent refractive indices n eq x , and n eq z shown in Figure 5(d).

Figure 5: 
(a) Subwavelength grating slab with periodic core of period Λ and silicon thickness H = 220 nm. (b) Effective index of the fundamental TE slab mode propagating along the z-axis (



n

eff
x




${n}_{{\mathrm{eff}}_{x}}$



) and x-axis (



n

eff
z




${n}_{{\mathrm{eff}}_{z}}$



). (c) Mapping function between the effective index of the fundamental TE mode of an isotropic slab and the corresponding core material refractive index. (d) Synthesized refractive index tensor components for a periodic-core slab with period Λ.

Figure 5:

(a) Subwavelength grating slab with periodic core of period Λ and silicon thickness H = 220 nm. (b) Effective index of the fundamental TE slab mode propagating along the z-axis ( n eff x ) and x-axis ( n eff z ). (c) Mapping function between the effective index of the fundamental TE mode of an isotropic slab and the corresponding core material refractive index. (d) Synthesized refractive index tensor components for a periodic-core slab with period Λ.

These parameters are analogous to n and n from Section 2.1, i.e., the metamaterial can be modeled as

(11) ε s = [ n eq x 2 0 0 0 n eq x 2 0 0 0 n eq z 2 ] .

From Figure 4(b) it is apparent that the slab model is indeed more accurate than the laminar model in the vicinity of the Bragg threshold, yet at the expense of a higher complexity in the modeling stage. This improvement is also seen when higher-order modes are considered: while for the 50-nm pitch both laminar and slab models give virtually exact results [see Figure 4(c)], for the 220-nm pitch the error in the effective indices is below 3% for the slab model, compared to 6% for the laminar approximation [see Figure 4(d)]. Regarding the transverse subwavelength waveguide, the slab model does also provide benefits compared to the laminar model [see Figure 4(f)–(h)].

The homogenization technique described above can be also applied when structural parameters of the subwavelength structure gradually change in the transverse direction. The subwavelength waveguide works then as a GRIN metamaterial described by a tensor ε(x), that is locally defined by a structural apodization function. Figure 6(a) and (d) show two representative examples where the duty cycle changes along the transverse direction, being DC(x) the subwavelength apodization function [see Figure 6(b) and (e)] [32, 47]. In Section 4 we describe the design process of an ultracompact SWG-assisted GRIN-lens spot size converter.

Figure 6: 
GRIN metamaterial waveguides with variable duty cycle based on (a) longitudinal and (d) transverse SWG structures. (b), (e) Apodized duty cycle DC(x) of the subwavelength structures in (a) and (d), respectively. (c), (f) Homogeneous anisotropic metamaterial tensor components 



n

eq
x



x



${n}_{{\mathrm{eq}}_{x}}\left(x\right)$



 and 



n

eq
z



x



${n}_{{\mathrm{eq}}_{z}}\left(x\right)$



 of the subwavelength structures of (a) and (d), respectively.

Figure 6:

GRIN metamaterial waveguides with variable duty cycle based on (a) longitudinal and (d) transverse SWG structures. (b), (e) Apodized duty cycle DC(x) of the subwavelength structures in (a) and (d), respectively. (c), (f) Homogeneous anisotropic metamaterial tensor components n eq x x and n eq z x of the subwavelength structures of (a) and (d), respectively.

2.3 Controlling birefringence with tilted subwavelength waveguides

So far, we have discussed how the longitudinal and transverse subwavelength waveguides shown in Figure 4(a) and (e) can be modeled using an anisotropic metamaterial core defined by the permittivity tensor ε = diag [ n x x 2 , n y y 2 , n z z 2 ] . In anisotropic materials, the behavior of light waves depends on the orientation of the electric field components with respect to the optical axis of the material. Therefore, the idea of tilting the anisotropic subwavelength structure to control the light propagation through a waveguide naturally emerges. Figure 7(a) shows a schematic of a silicon SWG waveguide in which the core segments have been tilted by an angle θ. Under the validity conditions of the discussed laminar (Section 2.1) and slab (Section 2.2) models, the core material of the tilted SWG waveguide core can be approximated by the permittivity tensor ε ˜ [48]:

(12) ε ˜ = T 1 ε T = n x x 2 θ 0 n x z 2 θ 0 n y y 2 θ 0 n x z 2 θ 0 n z z 2 θ ,

where T is the rotation matrix in the xz plane and θ is the tilt angle of the SWG segments.

Figure 7: 
(a) Schematic representation of a tilted SWG waveguide. The period of the structure, Λ
z
 = Λ/cos(θ), is much shorter than the wavelength, λ, of the light propagating along the z-axis. (b) Effective index of the fundamental TE and TM modes of the tilted subwavelength waveguide and the anisotropic homogeneous waveguide.

Figure 7:

(a) Schematic representation of a tilted SWG waveguide. The period of the structure, Λ z = Λ/cos(θ), is much shorter than the wavelength, λ, of the light propagating along the z-axis. (b) Effective index of the fundamental TE and TM modes of the tilted subwavelength waveguide and the anisotropic homogeneous waveguide.

The different elements of the tensor are given by

(13) n ˜ x x 2 = n x x 2 cos 2 ( θ ) + n z z 2 sin 2 ( θ ) , n ˜ y y 2 = n y y 2 , n ˜ z z 2 = n x x 2 sin 2 ( θ ) + n z z 2 cos 2 ( θ ) , n ˜ x z 2 = ( n z z 2 n x x 2 ) cos ( θ ) sin ( θ ) .

Note that the n ˜ y y component of the tensor does not change with the tilt angle. Therefore, this structure enables direct control over modal birefringence, only by changing the tilt angle θ, while using a constant duty cycle. This is illustrated in Figure 7(b), where we study the effective index of the fundamental TE and TM modes of the longitudinal SWG waveguide studied in Figure 4 with the tilt angle, using the slab model from Section 2.2. It is observed that the fundamental TE mode is strongly affected by the rotation angle, while the TM is virtually unaffected. These results are accurately predicted by the anisotropic metamaterial described by the tensor in Eq. (12). This control over the birefringence makes tilted SWGs ideal structures for the design of polarization management devices.

3 SWG-based devices

This section provides a thorough review of the state of the art in the application of subwavelength structures to integrated optical devices. We give insight into the operation of each of the devices and provide a detailed comparison of their performance. This article gathers the state of the art from 2018 onward, thereby complementing our previous reviews [18, 40, 41].

3.1 Ultralow loss waveguide crossings and bends

Low-loss and low-crosstalk waveguide bends and crossings are essential structures needed for dense integration. The main problem in waveguide bends is the optical length difference between the inner and the outer edges of the waveguide. This difference not only causes a mode mismatch between the modes of the straight and the curved waveguides but also produces intermode crosstalk. Subwavelength grating metamaterials have demonstrated to solve both problems by compensating the optical path difference via refractive index engineering. This compensation can be done using SWG structures with trapezoidal shape inside the curve [49, 50] [See Figure 8(a) and (b)] or tapered before and after the curve [51] [See Figure 8(c)]. The reduction on the bending loss of trapezoidal SWG allows to increase the Q-factor of SWG-based ring resonators by a factor between 3 and 5, compared to non-trapezoidal SWG bends [49]. Moreover, the mode shape adaptation allowed by SWG structures has enabled waveguide bends supporting up to four modes with a crosstalk lower than −20 dB, more than 10 dB lower than conventional multimode bends. In a non-optimized waveguide crossing, high insertion loss and crosstalk are a result of the transition between high-index-contrast waveguides, in which the mode field is highly confined, and the intersection zone, with no index contrast (free space). This discontinuity generates back-reflections and scattering into the perpendicular waveguides. Subwavelength grating structures reduce the index contrast, thereby smoothing the transition between the waveguides and the intersection zone and reducing the insertion loss and crosstalk (IL < 0.02 dB/crossing, CT < −40 dB) [52]. Graded-index devices [53], such as Luneberg lenses [54], constitute another interesting approach to design crossings. While these components have been demonstrated using periodic nanorods, the use of subwavelength-grating structures as those covered in this review paper would help shorten the device length by harnessing their anisotropic properties [32]. Implementing such GRIN material without SWG requires complex nonstandard fabrication method such as grayscale lithography [55].

Figure 8: 
(a), (b) Trapezoidal SWG structures inside the bend and (c) tapered SWG structures before and after the bend to compensate the lateral shift of the mode. Reproduced with permission from Refs. [49], [50], [51].

Figure 8:

(a), (b) Trapezoidal SWG structures inside the bend and (c) tapered SWG structures before and after the bend to compensate the lateral shift of the mode. Reproduced with permission from Refs. [49], [50], [51].

3.2 Broadband beam splitters and phase shifters

On-chip beam splitters are fundamental building blocks for almost any photonic integrated circuit, where they are often used cascaded and combined with other devices to perform complex functions. As such, their performance, both in terms of insertion losses and imbalance, as well as phase error between the outputs, is crucial. Additionally, broad bandwidths or the ability to handle multiple modes are becoming increasingly important [56, 57]. The properties of subwavelength structures can be exploited in combination with a variety of beam splitter configurations, illustrated in Figure 9, to enhance their functionality. In the following, we will be focusing on 50/50 splitters, as these are arguably the most widely used. For detailed performance information about the different splitters, the reader is referred to Table 1.

Figure 9: 
Configurations of conventional beam splitters and scanning electron micrographs of exemplary subwavelength implementations: (a) Y-branch, (b) adiabatic coupler, (c) asymmetric directional coupler, and (d) multimode-interference coupler. These devices exploit subwavelength structures to achieve smaller size, better performance, or both. Micrographs in (a), (b), and (c) are reproduced with permission from Refs. [59, 63, 65].

Figure 9:

Configurations of conventional beam splitters and scanning electron micrographs of exemplary subwavelength implementations: (a) Y-branch, (b) adiabatic coupler, (c) asymmetric directional coupler, and (d) multimode-interference coupler. These devices exploit subwavelength structures to achieve smaller size, better performance, or both. Micrographs in (a), (b), and (c) are reproduced with permission from Refs. [59, 63, 65].

Table 1:

Performance comparison of recently demonstrated subwavelength-enhanced on-chip beam splitters. All devices operate at near-infrared wavelengths.

Ref. Configuration Pol. # modes Simulation Experiment Length
IL (dB) IB (dB) BW (nm) IL (dB) IB (dB) BW (nm) (μm)
[62] 2 × 2 adiabatic coupler TE only 1 0.2 1 500 0.1 0.3 185 35
[63] 2 × 2 adiabatic coupler TE only 1 0.2 1 600 0.2 0.3 100 65
[70] 1 × 2 inverse design TE and TM 1 0.5 80 1 80 2.4
[58] 1 × 2 Y-branch TE and TM 1 0.2 400 3
[67] 1 × 2 slotted MMI TE and TM 1 0.3 200 4
[59] 1 × 2 Y-branch TE only 4 0.1 100 0.5 40 29
[69] 2 × 2 thin film interference TE only 3 0.5 <1 415 315
[64] 2 × 2 symmetric DC TE only 1 0.6 0.6 100
[65] 2 × 2 asymmetric DC TE only 1 0.4 0.3 300 1 0.5 200 6
[30] 2 × 2 anisotropic MMI TE only 1 1 0.5 500 1 1 300 25
[60]a 1 × 2 adiabatic coupler TE and TM 1 0.1 <0.1 500 1 1 390 200
[61]a 2 × 2 adiabatic coupler TE only 1 0.1 0.4 400 0.5 0.1 145 31

  1. aNon-SWG devices, included for reference. IL, Insertion Loss; IB, Imbalance; BW, Bandwidth.

In a Y-branch [Figure 9(a)] the input waveguide is gradually split in two, so that the input mode, φ in, smoothly evolves into the even super-mode of the two output waveguides, φ even; consequently, Y-branches exhibit no imbalance. One of the main drawbacks of this configuration is the sharp tip of the gap between the output waveguides, which is challenging to fabricate and can lead to performance degradation. Using inversely tapered waveguides with subwavelength structures between them, an ultracompact, polarization insensitive Y-branch has been recently proposed in Ref. [58]. Another approach is to substitute the gap between the waveguides by an array of subwavelength holes that adiabatically increase in size [59] thus synthesizing a metamaterial with a gradually decreasing equivalent index. This enables the implementation of a Y-branch that can operate up to the fourth mode with the remarkable performance shown in Table 1.

Adiabatic couplers, shown schematically in Figure 9(b), consist of waveguides that are tapered in width, thereby smoothly transforming the mode of the wide input waveguide, φ in,1, into the even supermode at the output, φ even, and the mode of the narrow input waveguide, φ in,2, into the odd supermode at the output, φ odd. This adiabatic transformation is demonstrated with measured bandwidths of almost 400 nm for both polarizations in Ref. [60], without using subwavelength structures. Typically, rather long devices are required to ensure adiabaticity. One way to overcome this issue is using a bent geometry to suppress undesired coupling and achieve a measured imbalance of only 0.12 dB [61]. Subwavelength structures provide another alternative to achieve shorter structures, by either lowering the equivalent index of the waveguides, or increasing the equivalent index of the gap between them [62, 63]. Using both approaches, devices with excellent performance in bandwidths exceeding 100 nm have been demonstrated.

Directional couplers [Figure 9(c)] exchange power between adjacent waveguides via the constructive and destructive interference of their supermodes, so that the beat length L π = (λ 0/2)/(n eff,evenn eff,odd) dictates the device length. In conventional devices, this beat length is strongly wavelength-dependent, which ultimately limits the bandwidth. For symmetrical directional couplers with 50/50 splitting, we proposed in Ref. [64] the superposition of a longitudinal subwavelength structure in the coupling region, to engineer modal dispersion and flatten the beat length. For 50/50 splitting, a judiciously designed asymmetric directional coupler exhibits an intrinsically larger bandwidth, which, when combined with subwavelength dispersion engineering, results in very large bandwidth: a measured bandwidth of 200 nm with very good performance was demonstrated in Ref. [65].

In multimode interference (MMI) couplers [Figure 9(d)], images of the input field are formed at distances governed by the beat length. In Ref. [30] we showed that an MMI comprising a longitudinal subwavelength structure exhibits a wide bandwidth which is a direct consequence of the metamaterial anisotropy. For an MMI of width W MMI the beat length is approximately given by L π aniso = 4 W MMI / 3 ( 1 / λ 0 ) n 2 / n , and the wavelength dependence of the term n 2 / n can practically cancel the term 1 / λ 0 . On the other hand, MMIs with subwavelength holes arrayed along the propagation direction have been shown to reduce device length by a factor of two [66]. This concept has been extended to an ultrashort device capable of handling both TE and TM polarization [67].

More recently, it has been shown that material anisotropy can also be exploited in phase shifters to achieve ultrabroadband operation [68]. As an example, consider a conventional phase shifter composed of two waveguide arms of different widths as shown in Figure 10. The modes φ A and φ B propagating through these waveguides exhibit different effective indices and thus accumulate a differential phase shift Δθ = (2π/λ0)(n eff,An eff,B)L. This phase shift is, however, strongly wavelength dependent. In Ref. [68] it was shown that if wide subwavelength waveguides are used in both arms the phase shift is approximately given by Δ θ = ( π / 4 ) 1 / W B - 1 / W A λ 0 n / n 2 , and as in the case of the SWG MMI, the wavelength dependence of the term n / n 2 can be designed to cancel the term λ 0. An operational bandwidth of 400 nm was predicted by simulation, and a four-fold reduction of the phase error compared to a conventional phase shifter was experimentally demonstrated over a 145-nm bandwidth.

Figure 10: 
(a) Conventional phase shifters use waveguides of different widths to achieve a controlled phase shift Δθ between them. When these waveguides are substituted by subwavelength structures the operational bandwidth of the phase shifter can be extended by a factor of four.

Figure 10:

(a) Conventional phase shifters use waveguides of different widths to achieve a controlled phase shift Δθ between them. When these waveguides are substituted by subwavelength structures the operational bandwidth of the phase shifter can be extended by a factor of four.

Subwavelength structures have also been successfully used in less conventional device configurations. In Ref. [69], a splitter capable of operating with three modes in a bandwidth in excess of 400 nm is proposed by exploiting a subwavelength-engineered thin film effect. An ultracompact inverse design splitter that handles both TE and TM polarization has been demonstrated in Ref. [70], achieving subdecibel losses over an 80-nm bandwidth.

3.3 On-chip beam shaping

On-chip beam manipulation has multiple applications including feeding of surface grating couplers [13, 71], [72], [73], environmental monitoring and security [74], on-chip particle acceleration [75], or wavelength demultiplexing [76]. Spot size conversion is arguably the most common beam shaping operation. For instance, beam expansion is required to transfer light from narrow Si-wire interconnecting waveguides to wide surface grating couplers. This operation is traditionally tackled by means of adiabatic transitions that smoothly change the width of the waveguide. This approach, however, requires a large footprint to ensure a virtually lossless adiabatic transition. Recently, several SWG-engineered devices that achieve efficient beam expansion in a compact footprint have been reported. Most of them implement a planar lens by either using subwavelength structures to realize a GRIN waveguide or tailoring the local phase front of an incident beam. In Ref. [32], a beam expander has been proposed using a wide SWG-based GRIN waveguide in which the duty cycle varies transversally to synthesize a parabolic index profile [see Figure 11(a)]. This GRIN waveguide behaves like a conventional lens that generates a collimated and expanded image of the incident beam at the focal length. The fabricated device achieves wideband (BW > 130 nm) and efficient (IL < 1 dB) 30-fold beam expansion in a tenth of the footprint of an optimized adiabatic taper that exhibits similar loss [77]. The design process of this SWG-based GRIN lens will be covered in Section 4 using the anisotropic properties of SWGs. A 4-f-system beam expander for multiple modes has been demonstrated in Ref. [78] by combining two SWG-based lenses, as shown in Figure 11(b). The lenses were implemented by varying the width of longitudinal silicon strips, achieving compact and efficient 5-fold expansion even for arbitrary field profiles. An insertion loss of ∼1.5 dB was experimentally shown, while the back-to-back inter-modal crosstalk of two devices was lower than −7 dB in a bandwidth of 25 nm.

Figure 11: 
On-chip integrated lenses implemented with SWG structures.
(a) GRIN-lens spot size converter [32]. (b) 4-f system comprising two GRIN lenses using a subwavelength stacking of silicon strips [78]. (c) On-chip lens using a subwavelength single-layer high-contrast transmit array metasurface [79]. Reproduced with permission from Refs. [32, 78, 79].

Figure 11:

On-chip integrated lenses implemented with SWG structures.

(a) GRIN-lens spot size converter [32]. (b) 4-f system comprising two GRIN lenses using a subwavelength stacking of silicon strips [78]. (c) On-chip lens using a subwavelength single-layer high-contrast transmit array metasurface [79]. Reproduced with permission from Refs. [32, 78, 79].

A different way to achieve beam focusing relies on metasurface-like phase-front shaping. In Ref. [79], subwavelength patterning of slots in a silicon slab was successfully used to locally control the phase front of a collimated incident beam, so that the fundamental mode of a wide waveguide (11 μm) is focused on to a narrow waveguide (500 nm) in a short length of 13.7 μm. The demonstrated device, shown in Figure 11(c), exhibited low loss (IL < 1 dB) over a broad bandwidth (BW > 200 nm).

The beam shape can also be controlled by means of on-chip deflectors. A deflector is a grating waveguide that progressively diffracts the guided light towards a slab, thereby maintaining the light confined within the chip plane, as shown in Figure 12. By properly apodizing the grating strength, any arbitrary field profile can be formed in the slab region, while the phase front can be tuned by chirping the period of the structure. This concept has been exploited in Ref. [80], using an SWG slab to mitigate the inherent off-chip radiation of diffractive gratings. The deflector achieved an extreme beam expansion (×70) from the fundamental mode of a standard silicon wire to a collimated Gaussian beam in a compact footprint of 100 μm × 5 μm. By judicious grating pitch design and SWG index engineering, off-chip radiation from the deflector can be completely eliminated [81]. To achieve a similar effect without SWG structures it is necessary to use either a third material with a refractive index ∼2.5 or multiple etch depths. Both strategies greatly complicate the fabrication compared with the SWG-based slab.

Figure 12: 
(a) SEM image of a fabricated sidewall grating deflector. The red arrows indicate the deflected light propagation direction and the green region comprises the SWG. (b) Field propagation inside the deflector. The light entering from the left gets progressively diffracted towards the wide waveguide as a collimated Gaussian beam.

Figure 12:

(a) SEM image of a fabricated sidewall grating deflector. The red arrows indicate the deflected light propagation direction and the green region comprises the SWG. (b) Field propagation inside the deflector. The light entering from the left gets progressively diffracted towards the wide waveguide as a collimated Gaussian beam.

3.4 Polarization management devices

Photonic integrated devices in silicon platforms often exhibit strong polarization dependence. This is due to the asymmetrical aspect ratio of the waveguide cross-section, the submicron scale of the waveguides, and the high index contrast of the materials. However, polarization independence is often desired because optical fibers randomly change the polarization state of the light to be injected into the chip. This can be achieved by using either polarization-diversity components – including polarization beam splitters (PBS), polarization rotators (PR), and polarization filters (PF) – or polarization-independent devices per se. Subwavelength engineering provides new degrees of freedom to tackle the design of both families of devices. It has to be highlighted that the operation of many of the devices discussed here only becomes possible as a result of the unique properties of subwavelength structures.

3.4.1 Polarization controlling devices

In the following, we discuss the latest SWG-based polarization management devices. They can be used for both polarization-diversity and polarization-division multiplexing (PDM) schemes. For a summarized perspective, we refer the reader to Table 2.

Table 2:

Performance comparison of recently demonstrated subwavelength-enhanced polarization beam splitters.

Ref. Device Operation principle Platform (Si thickness H) Length Simulation Experimental
Max. IL (dB)

@ BW (nm)
Min. ER (dB)

@BW (nm)
Max. IL (dB)

@ BW (nm)
Min. ER (dB)

@ BW (nm)
[82] PBS Bridged DC 340 nm 6 µm 0.63@100 20@60
[83] PBS DC 220 nm 7.2 µm 1@180 10@170 2.5@150 10@150
[84] PBS Tilted DC 220 nm 14 µm 1@100 20@86 1@143 15@72
[85] PBS Tilted DC 340 nm 7.5 µm 0.6@60 20@75
[86] PBS Skin-depth DC 300 nm*55 nm of SiO2 sandwiched in core 9.91 µm 0.4@100 20@50
[87] PBS Cascaded skin-depth DC 220 nm 130 µm 0.4@250 20@250
[88] PBS MMI 220 nm 133 µm 1.15@95 20@110 2.5@84 11.7@84
[89] PBS Tilted SWG MMI 220 nm 118 µm 1@131 20@131
[90] PBS Tilted SWG MMI 220 nm 118 µm 1.6@130 20@130 1@120 13@120
[31] PBS Skin-depth MMI 250 nm 12.3 µm 1@200 20@210 <1@200 >20@200
[92] PBS Mode evolution 340 nm 76.8 µm 0.3@270 20@270 1@230 20@240
[93] PR Hybridized structure 340 nm 4.8 µm 1@415 20@415
[94] PSR Mode evolution 220 nm 171 µm 0.5@120 15@120 1.4@90 15@105
[94] PSR Mode evolution 220 nm*90 nm shallow etched 185 µm 0.5@120 15@120 1.4@120 15@120
[95] PF Skin-depth cladding 250 nm 13 µm 1@415 25@415 1@415 20@415
[96] PF Birefringence engineering 250 nm 56 µm 1@210 36@210

In polarization beam splitters, horizontally and vertically polarized modes in the input waveguide are separated into two independent output ports [see Figure 13(a)]. A typical configuration is the asymmetric directional coupler, composed of two parallel waveguides with different optical properties. Subwavelength grating structures can be used to ensure that the phase matching condition is satisfied for a single polarization, so that only one of the input modes couples to the cross output. Multiple polarization-splitting asymmetric directional couplers have been recently demonstrated that use longitudinal [82, 83] and tilted SWG structures [84, 85]. The latter are especially interesting, as they can break the waveguide symmetry by simply tilting the core segments [see Figure 13(a.i)]. Another alternative to design directional coupler polarization beam splitters is by using skin depth engineering [33]. The evanescent field decay is controlled to isolate both arms of the directional coupler for the TE polarization, allowing the cross-coupling for the TM mode [86]. This device can be cascaded multiple times, improving their performance [87]. MMI couplers are also commonly used for polarization beam splitting. Compared with directional couplers, MMIs have larger bandwidths, tolerance to fabrication errors, and extinction ratio, yet at the expense of a larger footprint. In MMI devices, the images of the input fields are formed at different output ports [see Figure 13(a.ii)] for each polarization. SWG structures enable the birefringence engineering needed to tune the beat lengths for each polarization [88], [89], [90]. The birefringence engineering can be tuned either by tuning the period of the structure [88] or the tilt angle of the silicon segments [89, 90]. Dai et al. use SWG-enabled skin-depth engineering to achieve conventional MMI behavior for the TM polarization while the PBS acts like two uncoupled waveguides for the TE polarization [see Figure 13(a.iii)], experimentally demonstrating an outstanding performance, IL < 1 dB, ER > 20 dB over a bandwidth of 210 nm [31]. The operation of this device will be analyzed in more detail in Section 4. Furthermore, the control of SWG metamaterials over the effective indices of the modes and the polarization birefringence is especially convenient for mode-evolution devices [91]. Recently, mode-evolution-based polarization beam splitting has been reported with excellent performance [92].

Figure 13: 
Configurations of polarization controlling devices.
(a) Polarization beam splitters: a.i) Tilted directional coupler, reproduced with permission from Ref. [84]; a.ii) birefringence-engineered MMI, reproduced with permission from Ref. [88]; and a.iii) skin-depth-engineered MMI, reproduced with permission from Ref. [31]. (b) Polarization rotators and polarization beam splitter and rotators: b.i) Hybrid waveguide with refractive index and dispersion engineering, reproduced with permission from Ref. [93] and b.ii) mode-evolution splitter and rotator, reproduced with permission from Ref. [94]. (c) Polarization filters: c.i) Cladding anisotropy-engineered curve, reproduced with permission from Ref. [95] and c.ii) refractive-index-engineered waveguide, reproduced with permission from Ref. [96].

Figure 13:

Configurations of polarization controlling devices.

(a) Polarization beam splitters: a.i) Tilted directional coupler, reproduced with permission from Ref. [84]; a.ii) birefringence-engineered MMI, reproduced with permission from Ref. [88]; and a.iii) skin-depth-engineered MMI, reproduced with permission from Ref. [31]. (b) Polarization rotators and polarization beam splitter and rotators: b.i) Hybrid waveguide with refractive index and dispersion engineering, reproduced with permission from Ref. [93] and b.ii) mode-evolution splitter and rotator, reproduced with permission from Ref. [94]. (c) Polarization filters: c.i) Cladding anisotropy-engineered curve, reproduced with permission from Ref. [95] and c.ii) refractive-index-engineered waveguide, reproduced with permission from Ref. [96].

In polarization rotators, the input light is rotated into the orthogonal polarization state [see Figure 13(b)]. A polarization rotator (PR) and a polarization beam splitter (PBS) can be combined in a polarization beam splitter and rotator (PSR), which splits the input polarizations in two and then rotates the polarization state of one of outputs. Both PR and PSR are commonly implemented by exciting and beating hybrid modes or by using mode-evolution techniques. In hybrid-mode devices, the geometrical symmetry is broken [See Figure 13(b.i)] and the conversion length is typically highly wavelength-dependent. SWG structures enable broadband operation by engineering the metamaterial dispersion and flattening the conversion length response with the wavelength [93]. In mode-evolution devices, on the other hand, the effective index and polarization birefringence engineering helps design adiabatic transitions that gradually rotate the polarization states [See Figure 13(b.ii)] [94].

Polarization filters suppress only one of the input polarization components [see Figure 13(c)]. These devices are important because polarization beam splitters or rotators with ideal performance in terms of extinction ratio are not available, and pure polarization excitation is not achievable. Integrated polarization filters are based on the leakage or the reflection of the undesired polarization, while allowing the other to propagate. In Ref. [95] the authors add a transverse SWG structure to the outside lateral cladding of a bent waveguide core [See Figure 13(c.i)], the TE mode is laterally confined due to the skin depth effect, while the TM mode is completely radiated to the adjacent SWG metamaterial region. In Ref. [96], by leveraging the intrinsic birefringence of the silicon-on-insulator (SOI) platform and by engineering the refractive index, an SWG waveguide is designed that only supports TE modes [See Figure 13(c.ii)].

3.4.2 Polarization-independent devices

Subwavelength gratings enable the design of polarization-insensitive devices. There is not a generic design method to achieve polarization-insensitivity devices, but it depends on the particular device. In interference-based devices such as DCs or MMIs, SWG structures have enabled polarization independency by properly tuning the beat lengths of both polarizations (birefringence engineering) [67, 97], [98], [99]. In mode-evolution devices, SWG structures produce smooth transitions and can be designed for both polarizations simultaneously [58]. Likewise, in grating couplers, SWG segmentation can be used to properly adjust the radiation angle for both polarizations [100, 101]. Most of these designs rely on the chirping of the duty cycle of the periodic structure, affecting both polarizations simultaneously [see Figure 3(d)]. In this context, tilted SWG structures are auspicious topologies, as they mainly affect the TE polarization with virtually no effect on TM modes [see Figure 7(b)], thus facilitating the design of polarization-agnostic devices. To date, these advanced structures have been used to design a nonbirefringent waveguide [102]. However, we foresee an increase in the utilization of these subwavelength structures in more complex optical devices.

3.5 Narrowband spectral filtering devices

A Bragg grating is a periodic structure that works as a frequency-selective distributed mirror, providing a high reflectivity within a specific spectral band around the Bragg wavelength λ B, and transmitting the light at other wavelengths. In its basic form, an integrated Bragg reflector comprises a conventional waveguide with a periodic perturbation in either the composing material or the waveguide geometry. The reflection wavelength response of a canonical Bragg filter with period ΛB and length L = NΛB can be described using the following three parameters [103, Ch. 7]

  1. (i)

    Central wavelength:

(14) λ B = 2 n F Λ B ,

  1. (ii)

    Peak reflectance:

(15) | r ( λ B ) | 2 = | r | max 2 = tanh 2 ( κ L ) κ L 1 | r | max 2 1 , 

  1. (iii)

    Full bandwidth between nearest nulls:

(16) Δ λ = λ B 2 L n g 1 + ( κ L π ) 2 κ L 1 Δ λ λ B 2 π n g κ ,

where n F is the effective index of the Floquet–Bloch mode supported by the grating, n g is the group index of the unperturbed waveguide, and κ is the coupling coefficient between the forward and backward propagating modes. Since the coupling coefficient increases with the perturbation strength, Eq. (16) shows that Bragg filters with subnanometer bandwidth require weak perturbations.

An interesting approach to implement integrated Bragg filters is introducing a lateral corrugation on the waveguide sidewalls. Due to the high index contrast of the SOI platform, corrugations smaller than 10 nm are required to achieve bandwidths below 1 nm [104]. The need for such small corrugations in silicon waveguides hinders the practical implementation of narrowband, high-rejection SOI filters. Various narrowband integrated Bragg filters based on SWG waveguides have been proposed to overcome the drawbacks of conventional devices [28, 105], [106], [107], [108], [109], [110], [111], [112], [113]. In all cases, the filter is composed of a basic cell, of length ΛB, that is divided into two SWG sub-periods of lengths Λ1 and Λ2 = ΛB − Λ1. The desired weak perturbation is induced in the structural configuration of the second sub-period.

Figure 14(a) schematically shows the SWG-assisted Bragg grating topology proposed by Wang et al. [105, 106]. In this case, Λ1 = Λ2 = 0.5·ΛB, and the only difference between the sub-periods is the duty cycle (DC1 = a 11 ≠ DC2 = a 22). The authors report a measured 3-dB bandwidth of ∼0.5 nm and a peak reflectivity of 94.4% for a filter length of 1.12 mm [106].

Figure 14: 
SWG-based Bragg filter geometries. The Bragg period ΛB is divided in two slightly different subwavelength subperiods: Λ1 and Λ2 = ΛB − Λ1.
(a) Bragg filter assisted by a SWG with different duty cycles (Λ1 = Λ2, a
1 ≠ a
2) [105,106], (b) SWG-based double-width-corrugation Bragg filter, DWCBF (Λ1 = Λ2, ΔW ≠ 0) [107], (c) Symmetric-corrugation shifted Bragg filter geometry, CSBF (Λ1 ≠ Λ2) [108], (d) Asymmetric-corrugation shifted Bragg filter geometry, ACSBF (Λ1 ≠ Λ2, ξ ≠ 0) [108], (e) Multimode Bragg grating filter (Λ1 = Λ2, ΔW ≠ 0) [110], (f) Schematic representation indicating different parameters/degrees of freedom. Previous configurations (a–e) are particular realizations of this general case.

Figure 14:

SWG-based Bragg filter geometries. The Bragg period ΛB is divided in two slightly different subwavelength subperiods: Λ1 and Λ2 = ΛB − Λ1.

(a) Bragg filter assisted by a SWG with different duty cycles (Λ1 = Λ2, a 1a 2) [105,106], (b) SWG-based double-width-corrugation Bragg filter, DWCBF (Λ1 = Λ2, ΔW ≠ 0) [107], (c) Symmetric-corrugation shifted Bragg filter geometry, CSBF (Λ1 ≠ Λ2) [108], (d) Asymmetric-corrugation shifted Bragg filter geometry, ACSBF (Λ1 ≠ Λ2, ξ ≠ 0) [108], (e) Multimode Bragg grating filter (Λ1 = Λ2, ΔW ≠ 0) [110], (f) Schematic representation indicating different parameters/degrees of freedom. Previous configurations (a–e) are particular realizations of this general case.

In Ref. [107], Perez-Galacho et al. proposed an SWG-based Bragg filter with a small variation in the corrugation width of the perturbed subperiods [see Figure 14(b)]. The perturbation strength depends on the difference between the successive corrugation widths ΔW and not on their absolute values. Double-width corrugation Bragg filters (DWCBF) with a length of 1 mm, high rejection ratio exceeding 40 dB, and bandwidths of 1.1 nm were experimentally demonstrated, using the differential corrugation widths of ΔW ∼ 5 nm.

Improved robustness against fabrication errors and superior performance can be achieved by adding more degrees of freedom to the SWG subperiod perturbation, such as corrugation asymmetries and shifts. Oser et al. experimentally demonstrated that the combination of SWG waveguides with shifted asymmetric corrugations yields subnanometer bandwidths and large rejection ratios [108]. Specifically, the authors obtain Δλ = 0.8 nm with a rejection exceeding 40 dB for a symmetric-corrugation shifted Bragg filter (CSBF) [Figure 14(c)] and Δλ = 0.6 nm with a rejection level of 16 dB for an asymmetric-corrugation shifted Bragg filter (ACSBF) [Figure 14(d)], in both cases for a filter length of 1 mm.

Multimode Bragg gratings in which the fundamental mode is coupled to the backward first-order mode is yet another class of grating filters. Ultrahigh rejection levels (in excess of 80 dB) can be achieved by cascading several filter sections using single-mode S-bend waveguides. First-order-mode back-reflections are radiated in the S-bends without affecting the propagation of the forward fundamental mode, thereby enabling efficient noncoherent cascading of Bragg filters [109]. In Ref. [110], an SWG-based multimode Bragg grating filter with asymmetric half-period shifted corrugation as shown in Figure 14(e) was proposed. The asymmetry of the structure is designed to achieve an efficient coupling of the forward fundamental mode to the back-reflected first-order mode. By applying the noncoherent cascading concept, the authors have implemented a nine-section multistage filter with a measured rejection level of 60 dB and a bandwidth Δλ < 2 nm. A schematic representation including all degrees of freedom is shown in Figure 14(f). A high rejection (60 dB) third-order Bragg filter based on a Si membrane long-period diffraction-less grating core was proposed by Alonso–Ramos et al. [29]. In this work, authors design the grating core to suppress diffraction effects despite working with periods much larger than typically used in SWG structures.

A different approach to develop SWG-based Bragg filters with subnanometer spectral bandwidths was proposed by Ctyroky et al. [111]. The structure, illustrated in Figure 15(a), is composed of a conventional SWG metamaterial waveguide (period ΛSWG) loaded with an array of lateral silicon segments (period ΛB = 2ΛSWG) at a distance g from the SWG core. By engineering the refractive index, the modal field is delocalized from the SWG core to increase the overlap with the lateral loading segments, while maintaining the minimum feature size of the filter compatible with deep-ultraviolet (UV) lithography. The perturbation depends on the position of the lateral loading segments. The coupling coefficient κ exponentially decreases with the gap g because of the evanescent nature of the mode profile in the lateral cladding. Additionally, the longitudinal shifting ξ reduces the coupling coefficient as κ ( ξ ) = κ ( 0 ) cos ( π ξ / Λ B ) [114]. These two mechanisms provide an efficient control of the rejection band, enabling the design of SWG metamaterial-core Bragg filters with subnanometer spectral bandwidths and a minimum feature size of 100 nm [111]. Using this technique, Cheben et al. have reported measured 3-dB bandwidths as small as Δλ 3 dB ≈ 150 pm, for a filter length of 4.7 mm and TM polarization [28]. This configuration has been proposed to realize a high-sensitivity integrated sensor working at 1310-nm wavelength [115]. In Ref. [116], Sun et al. demonstrated a discretely tunable optical delay line that uses an SWG metamaterial core loaded by lateral segments with a periodic perturbation in the gap distance. The authors have recently shown a tilted SWG-waveguide Bragg grating with lateral loading segments [112]. By a judicious optimization of the tilting angle of the SWG blocks, a polarization-dependent tuning of the central wavelength is achieved.

Figure 15: 
(a) Bragg filter with SWG metamaterial core and lateral silicon loading segments [28, 111], (b) Complex Bragg filter with arbitrary spectral response, composed of an SWG core with apodized lateral silicon loading segments [113].

Figure 15:

(a) Bragg filter with SWG metamaterial core and lateral silicon loading segments [28, 111], (b) Complex Bragg filter with arbitrary spectral response, composed of an SWG core with apodized lateral silicon loading segments [113].

Complex Bragg filters with arbitrary spectral responses can be developed using a conventional SWG core with apodized lateral loading segments, as shown in Figure 15(b). In this filter configuration, the gap of each period, g i , is modulated to produce a specific coupling coefficient κ i , which is previously calculated from the target response by applying the layer-peeling algorithm [117]. The Bragg period ΛB,i is slightly adjusted to synthesize the phase profile of the target impulse response, as well as to counteract the variations of the self coupling term caused by the lateral loading segments. Pereira-Martin et al. have validated this approach by designing an astrophotonic filter comprising 20 nonuniformly spaced narrow notches matching atmospheric OH radical emission lines near 1550-nm [113]. Two implementations of this filter were experimentally evaluated, using a homogeneous silicon waveguide core, which exhibits measured 3-dB linewidths within the range 220–470 pm for a filter length of 3.6 mm; and an SWG metamaterial waveguide core, with measured 3-dB linewidths of 210–340 pm for a filter length of 6.4 mm.

3.6 High-performance devices for wavelength and mode division multiplexing

To increase the aggregated data rate of optical links, various multiplexing schemes that enable the superposition of independent data channels in the same waveguide have been proposed. Recently, researchers have shown an increased interest in multiplexing techniques such as wavelength division multiplexing (WDM), mode division multiplexing (MDM), or PDM, among others [118]. In this section we will focus on WDM and MDM devices, as PDM devices were already reviewed in Section 3.4. A key building block for these techniques is the demultiplexer, a device that can spatially separate the different channels into different waveguides. In the last years, several researcher groups have reported WDM and MDM devices that benefit from SWG index and dispersion engineering to achieve a superior performance compared to conventional devices.

3.6.1 Wavelength-division multiplexing devices

SWG structures have been applied mainly to contradirectional couplers (contra DCs). Contra DCs comprise two asymmetric waveguides that are placed in proximity (see Figure 16). The forward propagating mode in one waveguide is phase matched to the backward propagating mode in the other waveguide through the −1 harmonic that is induced by a periodic perturbation. This is achieved for n 1 + n 2 = λ 0/Λ, where Λ is the perturbation period and n 1,2 are the effective indices of the two waveguides [119]. Unlike Bragg gratings, in which the reflected light is coupled back into the same waveguide, contra-DCs evade the need for circulators, which are difficult to implement on-chip in integrated photonics [120]. Moreover, contra-DCs exhibit low losses, virtually infinite free spectral range, and box-like channel passband, which are all characteristics advantageous for WDM systems. Conventional contra-DCs formed by strip waveguides suffer from weak coupling, which results in long devices and narrow bandwidth. SWG waveguides have been used in contra-DCs to engineer the coupling strength between the waveguides, in order to reduce the device footprint or tailor channel bandwidth [121], [122], [123], [124], [125], [126]. In Refs. [121, 122], a strip waveguide is combined with an SWG waveguide to enhance the waveguide asymmetry while increasing the coupling strength, thus broadening the bandwidth, and reducing the device length. Another approach to attain the required asymmetry is to alternately offset the silicon blocks of both SWG waveguides [124]. In this work, the large mode delocalization of SWG waveguides resulted in an ultrawide channel bandwidth, as well as exceptionally low losses and low side-lobe suppression ratio (SLSR). Besides, contra-DCs allow for easy tuning, as their compact size provides efficient heating. This is shown in Ref. [126], where two independently controlled, thermally tuned contra-DCs are cascaded to develop an add-drop filter with tailorable bandwidth and central wavelength. In Ref. [125] the authors demonstrated that SWG-enabled contra-DCs are readily fabricable by deep-UV lithography. Table 3 summarizes the main performance metrics of recently reported contra-DCs. SWG engineering has also been applied, yet to a lesser extent, to other WDM devices. As shown in Ref. [127], SWG index engineering can be advantageously used to suppress most resonant modes of a ring-resonator-based demultiplexer, thus enlarging its free spectral range compared with conventional ring resonators. Except for that work, the use of SWG metamaterials in ring-resonator configurations has been limited mainly to sensing applications (Section 3.8). Another device that benefits from SWG metamaterials for WDM applications is the asymmetric directional coupler (ADC). In Ref. [128], an ADC was utilized to develop a broadband demultiplexer for the O and L bands, achieving excellent performance (1 dB insertion loss and crosstalk lower than −15 dB) and a short device length compared to conventional non-SWG devices. Surface grating couplers as wavelength demultiplexers (e.g., [101]) will be reviewed in more detail in Section 3.7.

Figure 16: 
Schematic of a WDM demultiplexer based on a contra DC. Here we show three different contra DCs, each tuned at a different wavelength (λ
1, λ
2, and λ
3). The signal centerd around λ
1 enters the demultiplexers and is extracted by the first contra DC (blue signal path). The typical transmission spectrum from the input port to the through port of the first device is illustrated (Drop operation). The signal centerd around λ
2 is injected from the Add port of the second device into the bus waveguide. A schematic transmission response for this Add operation is also shown. The inset shows the SEM image of the contra DC reported in Ref. [124] (reproduced with permission from Ref. [124]).

Figure 16:

Schematic of a WDM demultiplexer based on a contra DC. Here we show three different contra DCs, each tuned at a different wavelength (λ 1, λ 2, and λ 3). The signal centerd around λ 1 enters the demultiplexers and is extracted by the first contra DC (blue signal path). The typical transmission spectrum from the input port to the through port of the first device is illustrated (Drop operation). The signal centerd around λ 2 is injected from the Add port of the second device into the bus waveguide. A schematic transmission response for this Add operation is also shown. The inset shows the SEM image of the contra DC reported in Ref. [124] (reproduced with permission from Ref. [124]).

Table 3:

Performance comparison of recently demonstrated subwavelength-enhanced WDM (de)multiplexers based on contra DC.

Ref. IL@λ 0 (dB) 3-dB BW (nm) SLSR (dB) ER (dB) Notes
[121] <1.8 6.7 <−20 <−20
[122] 33.4 −2 −35
[123] <1 7–11 <−12
[124] 0.26 32.6 <−19 <−40
[125] 1.2 5 <−7 −30 O-band
[126] 2 6 −25 Tunable

  1. SLSR, Side Lobe Suppression Ratio; IL, Insertion Loss; IB, Imbalance; BW, Bandwidth; ER, Extinction Ratio.

3.6.2 Mode-division multiplexing devices

MDM demultiplexers spatially separate the optical signals carried by different modes propagating in a multimode waveguide into various single-mode waveguides. Several MDM demultiplexing architectures have been explored, including asymmetric Y-junctions, asymmetric directional couplers (ADC) [129], [130], [131], contra-directional couplers [132], or adiabatic tapers [133] (see Figure 17). In the following we review how SWG metamaterials have been used to improve the multiplexing performance of some of these structures.

Figure 17: 
Schematics of typical mode multiplexing architectures that have benefited from SWG engineering: (a) Adiabatic taper, (b) asymmetric DC, and (c) contra DC. The input ports (In 0, 1, 2) support the fundamental mode (φ
0) that the demultiplexer converts to one of the high-order modes (φ
1,2).

Figure 17:

Schematics of typical mode multiplexing architectures that have benefited from SWG engineering: (a) Adiabatic taper, (b) asymmetric DC, and (c) contra DC. The input ports (In 0, 1, 2) support the fundamental mode (φ 0) that the demultiplexer converts to one of the high-order modes (φ 1,2).

Adiabatic tapers are simple planar components with extremely broadband behavior that gradually transform an input mode into a target high-order mode [see Figure 16(a)]. These advantageous properties make them ideal building blocks for MDM demultiplexers. However, in order to ensure adiabaticity, conventional tapers need to be quite long (>100 μm) [134]. SWG metamaterials have been used to enhance the coupling strength and hence shorten the device length (∼25 μm) [133].

An important drawback of adiabatic tapers is that they are limited to few-mode conversion. For many-mode demultiplexing, the preferred architecture is the asymmetric directional coupler, which can be readily cascaded in series along a bus waveguide to target different high-order modes. ADC-based mode demultiplexers are generally formed by a multi-mode waveguide and a single-mode waveguide that are placed parallelly and in proximity, as illustrated in Figure 17(b). The fundamental mode of the single-mode waveguide is phase matched to one of the high-order modes of the multi-mode waveguide. In this scenario, energy transfer between both waveguides is achieved provided the phase-matched modes have nonzero overlap integral [103]. However, when conventional waveguides are used, the phase-matching condition generally holds only for a narrow operational bandwidth and stringent fabrication tolerances. This is caused by the difference in the effective index change with the wavelength and geometry of both waveguides. This limitation has been successfully overcome by using SWG index and dispersion engineering in Ref. [129]. In this work, the SWG is designed to match the effective indices and their dependence on the waveguide width for both ADC arms at the same time. The authors experimentally demonstrated an 11-mode demultiplexer with crosstalk less than −15 dB and insertion loss less than 2.6 dB at the central wavelength. They also demonstrate that the fabricated device can tolerate waveguide width variations of ±20 nm.

Contra-directional couplers are not only relevant for wavelength division demultiplexing, but can also be used for mode division demultiplexing provided that the target modes are phase matched, as shown in Figure 17(c). In Ref. [132], the authors leverage SWG index engineering to develop contradirectional coupling between modes of different order and polarization, thereby attaining mode, wavelength, and polarization multiplexing functionalities in the same device.

It is also possible to achieve the mode-multiplexing functionality by an appropriate joint use of several fundamental building blocks. In Refs. [135, 136], the authors combine a broadband SWG MMI, a phase shifter, and a symmetric Y-junction to implement a compact two-mode demultiplexer that achieves a low measured insertion loss (0.55 dB) and a crosstalk less than −18 dB in an outstanding bandwidth of 120 nm in the C band. This broadband operation is a direct consequence of the ultra-wide bandwidth of the SWG MMI used compared with conventional MMI devices.

3.7 High-performance off-chip couplers

Off-chip couplers are the essential input/output interfaces in integrated photonics. They are primarily utilized for fiber-chip coupling, as planar silicon waveguides have sub-micrometric mode sizes that are much smaller than the mode field diameter of a standard (SMF-28) optical fiber [137]. Likewise, off-chip couplers have been recently used as optical nanoantennas for applications such as satellite optical communications and light detection and ranging (LiDAR), which require shaping and steering of the radiated beams [138]. Edge couplers and surface grating couplers are the most important off-chip coupling devices (see Figure 18). While the former are in-plane spot size converters at the chip facet, the latter enable vertical off-chip radiation and can be placed anywhere on the chip, thereby facilitating wafer-scale testing. A comprehensive review of the fundamental coupling strategies in silicon photonics circuits can be found in Ref. [139].

Figure 18: 
Schematics of SWG-based fiber-chip off-chip couplers. (a) Edge coupler with a subwavelength adiabatic transition.
(b) Surface grating coupler with transverse subwavelength metamaterial patterning to apodize the radiated field profile.

Figure 18:

Schematics of SWG-based fiber-chip off-chip couplers. (a) Edge coupler with a subwavelength adiabatic transition.

(b) Surface grating coupler with transverse subwavelength metamaterial patterning to apodize the radiated field profile.

3.7.1 Fiber-chip edge couplers

Subwavelength segmentation in edge couplers has been widely adopted since its early demonstration and subsequent refinement by Cheben et al. [8, 25, 140]. SWG metamaterial edge couplers comprise a subwavelength-patterned waveguide with a smooth decrease of the duty cycle and the core width to adiabatically adapt the guided Floquet–Bloch mode to the fundamental mode of the optical fiber at the chip facet [see Figure 18(a)]. Unlike conventional inverse tapers [141], the subwavelength metamaterial tapers enable local adjustment of the equivalent refractive index of the waveguide core by simply changing the grating duty cycle. This additional degree of freedom is exploited to achieve optimum matching between waveguide and fiber modes for both TE and TM polarization, resulting in a 90% coupling efficiency with negligible polarization dependent loss [25]. In Ref. [38], researchers at IBM and GLOBALFOUNDRIES optimized the SWG edge coupler concept for standard SMF-28 fibers by partially removing the silicon substrate to prevent leakage loss of the delocalized mode [142]. These couplers were fabricated in monolithic silicon photonics production lines and provide complete fiber self-alignment via precisely etched V-grooves, with better tolerances than solid-core inverse-taper couplers. Coupling efficiencies of −0.7 dB and −1.4 dB were reported for TE and TM polarizations, respectively, over a 1-dB bandwidth broader than 60 nm in the O band. Similar structures achieve back-reflections lower than −30 dB [39]. An alternative way to increase the misalignment tolerance of conventional solid-core edge couplers is the use of trident-shaped topologies [143]. Teng et al. enhanced the trident concept by SWG structuring to achieve polarization independency, with coupling losses ranging from 0.5 to 2 dB over C band [144]. A dual-trident SWG edge coupler has been demonstrated at 1550-nm wavelength and TE polarization, exhibiting a coupling loss lower than 2 dB in a bandwidth of 92 nm [145].

3.7.2 Fiber-chip surface grating couplers

First proposed by Halir et al. [13], transverse SWG structuring of surface grating couplers, illustrated in Figure 18(b), has become a customary strategy for radiation-strength apodization. Since then, SWG metamaterials have been used not only to improve the coupling efficiency of surface grating couplers, but also to enlarge the bandwidth [146], to provide relaxed alignment tolerances [147], polarization insensitivity [148], and wavelength multiplexing for WDM applications [149], among others. The importance of SWG structuring of surface grating couplers is reflected in the number of articles that have been recently published to propose novel and efficient design techniques, including neural networks [150], machine learning [151], and weak form calculation and transformation optics [152].

Benedikovic et al. [71] leveraged subwavelength-grating properties to control both the upward diffraction efficiency and the field profile, thereby optimizing the directionality and the overlap with the fiber mode, achieving a theoretical coupling efficiency of up to 95%. Kamandar Dezfouli et al. used the degree of freedom introduced by subwavelength metamaterials to make the feature size of a perfectly vertical 75%-efficient grating coupler compatible with deep-UV lithography [153]. With the same aim, SWG structures were used in a bidirectional grating coupler with perfectly vertical radiation, achieving a respectable coupling efficiency (72% theoretically, 66% experimentally) for a minimum feature size larger than 100 nm [154]. Polarization insensitivity has been demonstrated by overlapping two perfectly vertical bidirectional grating couplers, arranged orthogonally. The resulting two-dimensional grating can also be SWG-apodized to improve the coupling efficiency (64% theoretically, 56% experimentally) [155]. As in the near-infrared telecom band, SWG apodization has proven useful also at mid-infrared wavelengths, where relaxed fabrication tolerances favor the utilization of subwavelength structures [156].

It is possible to broaden the bandwidth of surface grating couplers by reducing the group index of the grating [146]. This concept was exploited in Ref. [100] by leveraging SWG index engineering, obtaining simulated 3-dB bandwidths of 105 and 121 nm for TE and TM polarizations, respectively, yet at the expense of relatively low coupling efficiency. In Ref. [26] a different strategy to break the coupling efficiency–bandwidth trade-off was followed. The intrinsically narrowband nature of surface grating couplers was circumvented by using an SWG with zero-order radiation via a tilted silicon prism, yielding an unprecedented coupling efficiency–bandwidth product in excess of 75 nm (measured coupling efficiency of >80%, 1-dB bandwidth of 94 nm) [73].

The degrees of freedom introduced by SWGs can also facilitate the design of dual-band surface grating couplers, which handle light at two separate wavelengths, e.g. for WDM applications. In Ref. [149], the authors proposed a design strategy to tailor the separation of the coupling peaks and demonstrated dual-band focusing SWG grating couplers for a suspended 340-nm-thick SOI platform, with modest coupling efficiencies of 15–20% at central wavelengths in the range 1580–1660 nm. Following the same design flow and increasing the silicon thickness to 500 nm, the authors extended the operation to both near- and mid-infrared wavelengths simultaneously [157]. The device proposed in Ref. [101] couples the light from the optical fiber into two opposite output ports depending on the wavelength, thereby enabling demultiplexing at 1310 and 1550 nm. SWG sections are used to provide polarization independence with a minimum feature size of 100 nm, yielding coupling efficiencies around 35%. A similar approach has been recently shown to implement two-dimensional grating-coupling mode multiplexers for MDM [158].

3.7.3 Optical antennas

In the last few years, surface gratings couplers are being investigated as optical antennas. Embedded in optical phased arrays (OPAs), they are optimized to provide out-of-chip beam shaping and steering, which is required for LiDAR and free-space optical communications. In one-dimensional OPAs, the diffracted beam is steered in the two spatial directions using phase shifters (azimuthal direction) and wavelength scanning (elevation direction) [159]. Millimeter-long surface grating couplers can be utilized as optical antennas in 1D OPAs. By increasing the period of the loading segments, we have recently adapted the SWG-based Bragg-filter topology in Ref. [28] [see Figure 15(a)] to enable weak radiation [160], obtaining an experimental beam width of 0.1° and a wavelength sensitivity of 0.13°/nm, record metrics for deep-UV-compatible silicon-based optical antennas [161]. On the other hand, two-dimensional OPAs comprise a matrix of very short, densely packed antennas with strong radiation. In this case, phase shifters are used to steer the formed off-chip beam (in both azimuthal and elevation directions). An SWG-engineered deep-UV-compatible micro antenna concept with an impressive directionality of 94% and a 1-dB bandwidth of 230 nm was proposed by Khajavi et al. [162].

3.8 Optimized sensing waveguides

Silicon photonics is consolidating as one of the most promising technologies for developing cost-effective and portable lab-on-chip systems. Silicon-based waveguide sensors can detect the presence and concentration of biological and chemical species deposited onto the surface of the chip with high sensitivity and selectivity, which places silicon photonics in an advantageous position compared with alternative platforms. This opens a wide spectrum of applications, ranging from point-of-care clinical diagnosis to environmental monitoring of greenhouse gases to food safety control, to name a few [163].

The operation principle of most photonic integrated sensors relies on the interaction between the evanescent tail of a guided mode and the target analyte, which is typically in the upper cladding of the waveguide to facilitate the flow of dissolutions. The sensing waveguides are embedded in sensing architectures to transform any change in the concentration of the analyte into a change in the effective index of the mode and then into a quantity that can be easily measured. For example, effective index variations are converted to power variations in interferometric architectures – e.g., Mach–Zehnder interferometers (MZIs) – and to resonance wavelength shifts in resonant architectures – e.g., ring resonators and photonic crystal (PhC) cavities [164]. Depending on the required specificity, two types of sensing approaches are possible: Surface and bulk. The former requires prior functionalization of the top and sidewalls surfaces of the waveguide with selective bioreceptors, to which the target molecules are bonded producing an increase of the thickness of the adsorbed layer ( S d,surf = λ resonance / t adsorbed layer ). On the other hand, bulk sensors detect changes in the cladding (analyte) refractive index ( S d,b = λ resonance / n cladding ) .

SWG waveguides have proven to be convenient for sensing, because of their capability to increase the light-matter interaction [165], [166], [167]. Not only does the subwavelength segmentation delocalize the mode and thus increase the interaction volume, but it also creates a high field intensity between the silicon blocks, where the analyte is located. Using TE polarization, bulk waveguide sensitivities ( S w,b = n eff / n cladding ) of up to 0.8 (six-fold improvement compared with solid Si-core waveguide) and bulk device sensitivity (S d,b) above 400 nm/RIU have been achieved [165, 167, 168].

Over the past few years, research in SWG-based photonic integrated sensors has mainly been focused on four research directions. The first is the optimization of the waveguide dimensions. They should be carefully optimized to maximize sensitivity, while maintaining negligible leakage loss to substrate and feasible feature sizes. In Ref. [168], the bulk and surface sensitivities of SWG waveguides were calculated for a wide range of geometrical parameters (thicknesses, widths, and duty cycles), showing that silicon blocks that are narrower and thicker than those typically used in telecom applications (i.e., W = 450 nm, H = 220 nm) can provide bulk waveguide sensitivities close to unity. These results were confirmed by Milvich et al. by studying the surface sensitivity of various waveguide structures, including slot, double-slot, and conventional solid core waveguides, as well as different integration platforms (silicon nitride and SOI) and polarizations (TE and TM) [169]. Furthermore, in Ref. [170], the authors introduce a new parameter, the interaction efficiency, to quantify the probability of overlap between randomly distributed analyte molecules and the cladding mode field for different subwavelength waveguide geometries, thereby providing insight into the types of waveguides that yield more consistent and reliable sensing at low analyte concentrations. In Ref. [171], the authors thoroughly evaluate the trade-off between the mode confinement and the roughness-induced scattering loss of the most used sensing waveguides, including SWG waveguides.

The second important research direction includes investigation of new variants of SWG waveguides to further enhance sensing capabilities. The variants can be grouped in two different categories: (i) Multibox or multislot SWG waveguides and (ii) substrate over-etched SWG waveguides. The former combines the slot waveguide and the SWG waveguide by patterning the silicon segments in both the transverse and longitudinal directions [Figure 19(a) and (b)]. In addition to introducing more parameters to control the mode delocalization, this geometry provides an enlarged binding surface around each silicon block and an increased field intensity for the TE mode in the gaps [see the inset of Figure 19(b)], which significantly enhances the surface sensitivity. Experimental results were obtained in Ref. [172] for one slot [Figure 19(a)] and in Ref. [173] for four slots [Figure 19(b)], showing a two-fold enhancement in the surface sensitivity compared to the nominal SWG waveguide (2000 pm/nm vs 1000 pm/nm). The drawback of multi-box waveguides is that scattering losses are higher than in conventional strip and SWG waveguides, so that the improvement in sensitivity does not translate into an improvement in the intrinsic limit of detection, iLoD = λ o / Q · S d , a figure of merit defined in Ref. [174]. On the other hand, in substrate over-etched SWG waveguides, the light–matter interaction can be enhanced by partially removing the buried oxide underneath to reduce the vertical asymmetry. Depending on the isotropy of the wet etching, two different types of waveguides have been obtained: The pedestal-SWG [175], shown in Figure 19(c), and the substrate over-etched-SWG [176], shown in Figure 19(d). In both cases, S w , b 1 , and S d , b of 545 nm/RIU and 575 nm/RIU are achieved, respectively, when the waveguides are embedded in a ring resonator architecture. The main drawback of the pedestal-SWG design is that, in order to guarantee the mechanical stability, the maximum over-etching depth [V in Figure 19(c)] is limited by the length of the silicon blocks [I in Figure 19(c)].

Figure 19: 
Summary of the most relevant recent works in SWG-based photonic integrated sensors.
(a) SEM image of the slot SWG waveguide (reproduced with permission from Ref. [172]). (b) SEM image of the multibox SWG waveguide embedded in a ring-resonator architecture. The inset shows the transverse electric field profile at the middle of the silicon blocks (reproduced with permission from Ref. [173]). (c) Schematic of the pedestal SWG waveguide (reproduced with permission from Ref. [175]). (d) Schematic of the SWG waveguide with over-etched substrate (reproduced with permission from Ref. [176]). (e) Schematic of the multibox SWG phase-shifted Bragg grating resonator, in which the external Si blocks of period Λ2 constitute the Bragg corrugation (reproduced with permission from Ref. [177]). (f) SEM image of the multislot photonic crystal cavity (reproduced with permission from Ref. [178]). (g) SEM image of the multislot SWG Bragg grating (reproduced with permission from Ref. [179]). (h) Schematic of the bimodal SWG interferometric sensor (reproduced with permission from Ref. [180]). (i) Schematic of the SWG-based Bragg filter topology used as a sensor (reproduced with permission from Ref. [115]).

Figure 19:

Summary of the most relevant recent works in SWG-based photonic integrated sensors.

(a) SEM image of the slot SWG waveguide (reproduced with permission from Ref. [172]). (b) SEM image of the multibox SWG waveguide embedded in a ring-resonator architecture. The inset shows the transverse electric field profile at the middle of the silicon blocks (reproduced with permission from Ref. [173]). (c) Schematic of the pedestal SWG waveguide (reproduced with permission from Ref. [175]). (d) Schematic of the SWG waveguide with over-etched substrate (reproduced with permission from Ref. [176]). (e) Schematic of the multibox SWG phase-shifted Bragg grating resonator, in which the external Si blocks of period Λ2 constitute the Bragg corrugation (reproduced with permission from Ref. [177]). (f) SEM image of the multislot photonic crystal cavity (reproduced with permission from Ref. [178]). (g) SEM image of the multislot SWG Bragg grating (reproduced with permission from Ref. [179]). (h) Schematic of the bimodal SWG interferometric sensor (reproduced with permission from Ref. [180]). (i) Schematic of the SWG-based Bragg filter topology used as a sensor (reproduced with permission from Ref. [115]).

The third line of research direction investigates new sensing architectures. Since the demonstration of the multibox SWG waveguide as a sensor [173], several works have tried to evaluate its performance in architectures other than ring resonators. For example, in Ref. [177], the multibox SWG waveguide is used in a phase-shifted Bragg grating (PSBG) resonator [Figure 19(e)]. Although the device sensitivity is almost the same as that of ring resonators, the absence of bends produces a smaller footprint (∼200 versus 3000 μm2) and a higher Q-factor (8000 versus 2600), which results in an improved iLoD (see Table 4). A similar performance was obtained in a photonic-crystal-based multibox SWG sensor in which a Bragg grating cavity is defined by gradually increasing the length of the silicon blocks from the center to the ends of the structure, as shown in Figure 19(f) [178]. In Ref. [179], the multislot SWG waveguide is designed to operate in the Bragg regime and, by using the first side lobe of the transmission spectrum as a sensing wavelength peak, a compact sensor with a length of only 9.5 μm and a sensitivity of 730 nm/RIU is experimentally demonstrated in a 340-nm-thick SOI platform [Figure 19(g)]. In relation to SWG-based interferometric architectures, some of the best performance has been achieved in Ref. [180]. Here, the concept of bimodal waveguide biosensor proposed in Ref. [181] is extended also to SWG waveguides, with promising experimental results (S d,b = 2270 nm/RIU) [see Figure 19(h)]. One-dimensional photonic crystal nanobeam cavities are also nano-periodic structures with an excellent potential for optical sensing applications – we refer the reader to a recent review paper on this topic [182].

Table 4:

Main performance metrics of SWG-based sensors.

Ref. Waveguide/Architecture S w,b (RIU/RIU) S d,b (nm/RIU) FSR (nm) iLoDb (×10−5 RIU)
[172] Slot SWG/MZI 0.79 −599 13 n/a
[173] Multislot SWG/Ring resonator 0.80 580 6 100
[175] Pedestal SWG/Ring resonator 1.03 545 13 157
[176] Substrate over-etched SWG/Ring resonator 0.97 575 5 175
[177] Multislot SWG/Phase shifted Bragg grating n/a 579 20 36
[178] Multislot SWG/PhC cavity 586 >70 70
[179] Multislot SWG/Bragg side lobe n/a 730 n/a n/a
[180] Bimodal SWG/MZI n/a 2270 ∼55 n/a
[115] SWG/Narrowband Bragg filter 0.93 507 FSR free 5.1

The fourth line of action focuses on reducing the waveguide propagation loss, which can largely improve the limit of detection [183]. Since most of the analytes of interest are dissolved in aqueous solutions and the optical absorption of water is the main limiting loss factor in the sensor system at 1550-nm wavelength ( α H 2 O = 52.5 dB/cm), increasing the waveguide sensitivity also increases the losses in the same proportion. This is the reason why all the aforementioned works did not significantly reduce the iLoD. Therefore, moving towards optical wavelengths with lower losses due to water absorption, like telecom O-band (λ 0 ∼1310 nm; α H 2 O = 6.6 dB/cm), is a promising alternative. Some initial theoretical investigations have been made in this direction [115]. We explored a sensing architecture based on an SWG waveguide that is evanescently coupled to periodic lateral loading segments [Figure 19(i)], a novel SWG-based topology that was initially proposed for ultra-narrow Bragg filters (see Section 3.5, Ref. [111]). The simulation results showed a bulk sensitivity of 507 nm/RIU, a Q-factor of 4.9 × 104 and a iLoD of 5.1 × 10−5 RIU, the latter yielding about one order of magnitude improvement compared to state-of-the-art SWG-based sensors. Table 4 summarizes the main performance metrics of the discussed works. We believe that SWG waveguides for silicon photonics integrated sensors can pave the way for the development of future lab-on-chip applications.

3.9 Suspended waveguides for mid-infrared sensing and communications

The mid-infrared band is especially attractive for sensing applications, because the chemical bonds of the molecules of many organic and inorganic compounds resonantly absorb photons in this wavelength range (λ = 2–20 µm). From Soref’s seminal paper [184], the interest in silicon photonics as a potential technology for integrated mid-infrared applications has steadily grown [185], [186], [187].

The standard SOI platform is not optimal for wavelengths above 4 µm, because of the prohibitive loss of the silicon dioxide layer [21]. Alternative materials have been proposed to cope with this inconvenience, including sapphire [188] and silicon nitride [189], but the operating range remains limited by the intrinsic losses of the lower cladding materials.

In suspended silicon membranes, the buried oxide (BOX) layer of the SOI wafer is removed by an acid solution delivered through a set of holes that are previously etched in the silicon layer [190, 191]. Suspended silicon membranes benefit from the entire transparency window of silicon, up to ∼8 µm. As shown in Figure 20(a), SWGs are often utilized to define a lateral cladding with a three-fold function: (i) mechanically support the waveguide core, (ii) allow the flow of the hydrofluoric (HF) acid to etch the BOX layer, and (iii) synthesize a cladding-core index contrast required for waveguiding [see Figure 20(b)]. The latter produces lateral confinement within the waveguide core by only using a single reactive-ion etch step, which is an obvious advantage compared to membranes that require two steps to define a rib [190]. The design of SWG-cladding suspended waveguides requires a judicious selection of dimensional parameters to find a sweet spot for different requirements including mechanical robustness, fabrication feasibility, and low-loss single-mode optical guiding. A comprehensive analysis of SWG suspended waveguides can be found in Ref. [191].

Figure 20: 
Schematics of (a) a suspended silicon waveguide with SWG lateral cladding and (b) its homogeneous model. SEM images of (c) the suspended silicon waveguide at 7.7-μm wavelength (reproduced with permission from Ref. [195]), (d) the suspended silicon slot waveguide at 2.2-μm wavelength (reproduced with permission from Ref. [197]), and (e) the suspended germanium waveguide at 7.7-μm wavelength.

Figure 20:

Schematics of (a) a suspended silicon waveguide with SWG lateral cladding and (b) its homogeneous model. SEM images of (c) the suspended silicon waveguide at 7.7-μm wavelength (reproduced with permission from Ref. [195]), (d) the suspended silicon slot waveguide at 2.2-μm wavelength (reproduced with permission from Ref. [197]), and (e) the suspended germanium waveguide at 7.7-μm wavelength.

Soler et al. reported a suspended silicon waveguide with lateral SWG cladding, at a wavelength of 3.8 μm [192]. Using a silicon thickness of 500 nm and a waveguide core width of 1.3 μm, a propagation loss as low as 0.82 dB/cm was achieved [193]. Other components such as MMI couplers and Mach–Zehnder interferometers were also implemented [193]. The first uncooled silicon waveguide-based bolometer was demonstrated for this suspended waveguide, with its surface being assisted by gold plasmonic antennas to absorb light and by amorphous silicon thermometers, yielding sensitivities up to 1.13 ± 0.04% change in resistance per milliwatt of input power [194].

The SWG suspended silicon waveguide concept was extended to λ 0 = 7.7 μm, the longest wavelength at which a silicon-based platform has yet been demonstrated. The silicon thickness was increased to 1.5 µm to minimize leakage losses toward the substrate. Similarly, a wider core of 2.9 µm was chosen to support the fundamental mode with low lateral leakage. An SEM image of this waveguide is shown in Figure 20(c). 90°- and S-bends with negligible losses were also reported [195].

In Ref. [196], the SWG-cladding suspended silicon platform was optimized for sensing at λ 0 = 2 μm. A commercial 340-nm-thick silicon layer was utilized for a trade-off between propagation losses and enhanced sensitivity to refractive index variations in the cladding. A quality factor Q of 15,300 was measured for a microracetrack resonator, for which a sensitivity of 337.5 nm/RIU was predicted. Zhou et al. developed a slot-waveguide suspended platform based on the same SWG-cladding geometry [197]. An SEM image of this structure is provided in Figure 20(d). Waveguides, curves, and racetrack resonators were demonstrated, yielding a propagation loss of 2.8 dB/cm, a bend loss of 0.15 dB per 90-degree curve, and an intrinsic optical Q factor of 12,600, respectively. These structures can potentially provide high sensitivity for applications up to a wavelength of 8 μm because of the high field concentration within the slot, enhancing the light interaction with the analyte.

At still longer wavelengths, λ 0 > 8 µm, the high absorption loss of silicon makes this material unsuitable for the guiding layer and germanium can be used instead. Recently, SWG-cladding suspended germanium waveguides with ∼5 dB/cm propagation loss have been confirmed at 7.7 µm as a proof of concept, for TE polarization [198] [see Figure 20(e)]. For this platform, a suspended germanium micro-antenna was proposed as an input/output coupler between the chip and a long-wave chalcogenide optical fiber, with a design coupling efficiency of ∼40%, a 1-dB bandwidth larger than 430 nm, and an angular bandwidth of ±10° [199]. This platform can potentially cover the entire transparency window of germanium, up to a wavelength of ∼15 µm.

While initially targeted for the mid-infrared band, SWG-cladding suspended waveguides can be advantageously used for other wavelength ranges and applications as well as in novel waveguide platforms. A silicon-carbide-based platform was theoretically proposed for nonlinear applications and quantum photonics [200]. Since the base wafer comprises a thin layer of silicon carbide (n SiC ∼ 2.6) on silicon (n Si ∼ 3.5), here the suspension provides the vertical confinement for light guiding, while the SWG lateral cladding makes the silicon substrate accessible in a single dry etch step.

4 A “hands-on” with advanced subwavelength structures

In Section 2 we discussed the modeling of SWG structures using anisotropic metamaterials and showed that these models can provide accurate results, comparable to rigorous simulations of SWG waveguides in terms of modal analysis (see Figure 4). Here we aim to illustrate how these models can be leveraged to simplify the design and simulation of complex subwavelength-based devices. Specifically, we focus on the design of a polarization beam splitter and a waveguide lens, simulate them using the homogenization tools described in Section 2, and compare the results with rigorous, but time-consuming, 3D-FDTD simulation of the device with subwavelength geometry.

4.1 Broadband polarization beam splitter

As a first example of advanced devices based on subwavelength structures, we consider the broadband polarization beam splitter reported in Ref. [31]. The geometry of the device is shown in Figure 21(a) and consists of two parallel waveguides composed of a longitudinal subwavelength structure, with a transverse subwavelength structure between them. The basic idea is that for TM polarization (along the y direction) the electric field is parallel to the interfaces in both the longitudinal and the transverse subwavelength structures. Thus, for TM polarization the device “looks” homogeneous and acts as an MMI coupler that forms an image of the input field at the cross port. For TE polarization (in the xz plane) the transverse subwavelength structure suppresses coupling between the parallel waveguides [33], so that light from the input propagates to the bar port. While full simulation of this structure requires time-consuming 3D-FDTD techniques, in the following we show how, using the metamaterial models discussed in Section 2, simple modal simulations can be used to analyze this device.

Figure 21: 
(a) A polarization splitter based on a dual subwavelength structure (reproduced with permission from Ref. [31]). (b) Using the anisotropic material models developed in Section 2, a much simpler, equivalent device can be constructed. (c, d) Electric field propagation obtained via modal simulation for TE and TM polarization through the equivalent device. (e) Power in the fundamental modes of the cross and bar ports as function of device length.

Figure 21:

(a) A polarization splitter based on a dual subwavelength structure (reproduced with permission from Ref. [31]). (b) Using the anisotropic material models developed in Section 2, a much simpler, equivalent device can be constructed. (c, d) Electric field propagation obtained via modal simulation for TE and TM polarization through the equivalent device. (e) Power in the fundamental modes of the cross and bar ports as function of device length.

The design is based on a subwavelength structure with a pitch of Λ = 250 nm. As discussed in Section 2, for this pitch the laminar model is usable, but the slab model yields very good results [see Figure 4]. We thus follow the steps outlined in Section 2.2 to obtain the equivalent metamaterial of the 250-nm-thick silicon layer, with a pitch of Λ = 250 nm, and a duty cycle of 74% and 70% for the longitudinal and transverse subwavelength structures, respectively. As a result, we obtain the diagonal index tensors for the equivalent structure shown in Figure 21(b). Note that the tensor for the transverse structure has been rotated by 90° using Eqs. (12) and (13), to account for the rotation in the coordinate system compared to Figures 3 and 4. From the refractive index tensors it is observed that indeed for TM polarization the device is virtually homogeneous, since the n yy component is very similar in all the device. On the other hand, for TE polarization, the large n zz component in the intermediate region enhances the field confinement inside that region [33]. The structure shown in Figure 21(b) can be readily analyzed with a mode propagation software [45], with significantly less computational effort than FDTD. The propagation of the electric field through the device for TE and TM polarization is shown in Figures 21(c) and 20(d), respectively. Notably, this comparatively simple model accurately predicts not only the polarization splitting behavior of the device, but also the optimal length: From Figure 21(e), obtained with the homogeneous model, the optimal device length is estimated in about 7.2 µm, which translates to 29 longitudinal periods, while the optimized value obtained in Ref. [31] via 3D-FDTD simulations of the subwavelength geometry is 30 periods.

4.2 Ultracompact GRIN-lens-based spot size converter

GRIN materials have demonstrated multiple applications in integrated photonics, including beam shaping [32, 72], waveguide bending [50, 51], waveguide crossing [53], or cloaking [201]. Subwavelength grating structures allow the synthesis of on-chip GRIN metamaterials [32, 47, 72] without requiring complex fabrication techniques such as grayscale lithography [55]. In this section we describe the design algorithm to synthesize a parabolic GRIN metamaterial. Then, we use the metamaterial to design a lens-based spot size converter, proposed in Ref. [32] and schematically shown in Figure 22(a). This device consists of a longitudinal SWG waveguide with constant period and an optimized duty cycle in the transverse direction, DC(x) = a(x)/Λ. For illustration purposes we assume an operating wavelength λ 0 = 1550 nm, a silicon thickness H = 220 nm, a lens pitch Λ = 240 nm, and a lens width w GRIN = 16 µm. The minimum gap at the center of the lens is set to 85 nm (DCmax ∼ 0.65) to ensure fabrication feasibility. The design procedure to calculate the duty cycle function, DC(x), and the homogeneous anisotropic model of the GRIN core, ε ( x ) = diag [ n x x 2 ( x ) , n y y 2 ( x ) , n z z 2 ( x ) ] , is summarized in Box 1.

Figure 22: 
(a) Schematic representation of the SWG GRIN lens proposed in Ref. [32]. (b) Desired gradual index profile (n
GRIN) and the corresponding n

zz
 implemented with the proposed structure. Inset: Duty cycle apodization function DC(x) which implements the desired profile. (c) Schematic representation of the GRIN metamaterial which models the proposed structure. (d, f) Full-vectorial 3D-FDTD simulation of the electric field (R{E

x
(x,0,z)}) propagating through a GRIN lens implemented with (d) a periodic SWG structure and (f) a gradual anisotropic metamaterial 



ε

(
x
)

=
d
i
a
g

[


n

x
x

2


(
x
)

,

n

y
y

2


(
x
)

,

n

z
z

2


(
x
)


]




$\boldsymbol{\varepsilon }(x)=diag[{n}_{xx}^{2}(x),{n}_{yy}^{2}(x),{n}_{zz}^{2}(x)]$



, with the indices shown in (b). (e) Metamaterial tensor components n

xx
 and n

zz
 as function the duty cycle DC = a/x.

Figure 22:

(a) Schematic representation of the SWG GRIN lens proposed in Ref. [32]. (b) Desired gradual index profile (n GRIN) and the corresponding n zz implemented with the proposed structure. Inset: Duty cycle apodization function DC(x) which implements the desired profile. (c) Schematic representation of the GRIN metamaterial which models the proposed structure. (d, f) Full-vectorial 3D-FDTD simulation of the electric field (R{E x (x,0,z)}) propagating through a GRIN lens implemented with (d) a periodic SWG structure and (f) a gradual anisotropic metamaterial ε ( x ) = d i a g [ n x x 2 ( x ) , n y y 2 ( x ) , n z z 2 ( x ) ] , with the indices shown in (b). (e) Metamaterial tensor components n xx and n zz as function the duty cycle DC = a/x.

Box 1

SWG metamaterial index design algorithm

  • ① Calculate the tensor components of the homogeneous material model as a function of the duty cycle (Section 2): { n xx ( D C ) n z z ( D C )

  • ② Impose the GRIN goal function: n xx ( x ) = n G R I N ( x )

  • ③ Calculate the shape of the silicon segments using the mapping function calculated in ①: n xx ( x ) n xx ( DC ) DC ( x )

  • ④ Complete the anisotropic model by determining the remaining GRIN metamaterial tensor components: { n y y ( x ) = n x x ( x ) DC ( x ) n z z ( DC ) n z z ( x )

Step ①: We start from the homogenized material model described in Section 2, which allows the calculation of n xx (DC) and n zz (DC), i.e., the permittivity tensor components of the metamaterial as a function of the duty cycle. Step ②: We enforce that n xx (x) follow the desired GRIN function, which, in this specific case, is the parabolic law n GRIN ( x ) = n 0 1 ( α x ) 2 , where n 0 is the refractive index at the center of the lens and α determines the curvature of the parabola [see Figure 22(b)]. Step ③: The shape of the silicon segments, DC(x), is calculated using n xx (DC) as a lookup table. In this example, since n xx (DC) is practically linear within the range DC ∼ 0.25–0.65 [see Figure 22(e)], the duty cycle DC(x) is also a nearly parabolic function. Step ④: The remaining GRIN metamaterial tensor components are determined, thereby completing the homogenization of the GRIN metamaterial.

To validate the proposed anisotropic model and the behavior of the SWG-based GRIN metamaterial, we compare the simulated field propagation through i) the periodic lens and ii) the homogeneous anisotropic lens defined by a gradual permittivity tensor ε ( x ) = diag [ n x x 2 ( x ) , n y y 2 ( x ) , n z z 2 ( x ) ] [see Figure 22(c)]. We excite the modes of the structures with a 1 × 0.22 μm2 Gaussian beam. As shown in Figure 22(d) and (f), in both cases, the simulated structures operate as parabolic GRIN lenses, expanding and collimating the field profile distributions at specific distances: f per ∼ 11 µm and f anis ∼ 12 µm. As expected, the homogeneous anisotropic model gives an accurate prediction of the device length (f anis f per), which clearly confirms the appropriateness of the proposed homogenization technique. Even more, the SWG-based GRIN lens is significantly shorter than the homogeneous isotropic lens, which is mainly caused by the inherent anisotropy of the device (i.e., n zz n xx ).

5 Conclusions and outlook

Since the invention of silicon SWG metamaterial waveguides at the beginning of the millennium [8], [9], [10], [11], [12], [13], [14], [15], [16], research on these structures has been growing rapidly, with many impressive devices demonstrated for a wide range of applications. Indeed, the flexibility afforded by SWG metamaterial waveguides enables designers to choose the optimum material properties for each application. This has led to the demonstration of integrated photonic devices with unprecedented performance, often covering multiple or even all optical communications bands at the same time or extending the operation range of silicon photonic devices into mid-infrared. Significant advances have also been made in fiber-to-chip coupling, with solutions that offer sub-decibel coupling loss and ultra-broad bandwidths, both using edge and grating couplers. Indeed, as more research groups incorporate subwavelength structures into their design flows, the discovery of new SWG-engineered devices is expected to continue and expand into new application fields such as quantum photonics.

Subwavelength structures require feature sizes of the order of about 100 nm for operation in the near-infrared band. Such small features can be readily defined with e-beam lithography, but for conventional wafer-scale deep-UV lithography techniques, they are often at the limit of what can be reliably reproduced. To make the benefits of on-chip SWG metamaterials available to the wider silicon photonics industry, it is pivotal to overcome this technological limitation. Deep-UV immersion lithography with 60 nm resolution is being offered by several silicon photonics foundries, and some of them are already using SWG technology in their Process Design Kits [38, 202], [203], [204]. On the other hand, an effort is also being made to design geometries that enable subwavelength operation with larger feature sizes. A recent example is the bricked SWG waveguide [205], allowing substantially larger minimum feature sizes compared to conventional subwavelength structures. The bricked pattern only requires a single etch step and makes use of a Manhattan-like geometry, with a uniform grid and pixel dimensions as large as 150 × 150 nm2 for telecom wavelengths. We believe that the combination of advanced lithography techniques and novel subwavelength geometries will pave the way towards ever increasing adoption of subwavelength engineered devices in mass scale fabrication processes.

Notably, fabrication related constraints are relaxed significantly in the surging field of mid-infrared photonics, since the feature sizes scale directly with wavelength. Subwavelength structures are thus expected to play a key role in this field in the near future. In the visible wavelength range, silicon is opaque, and while silicon nitride offers low loss propagation, the main challenge for waveguide subwavelength structures again lies in reliable fabrication. For silicon nitride waveguides at a wavelength of 600 nm, the feature size for a subwavelength structure is of the order of 60 nm. Thus, in the visible wavelength range, subwavelength structures will become of increasing interest using high resolution deep-UV immersion lithography.

Of course, the use of subwavelength structures need not be limited to any specific material platform or wavelength range. When considering alternatives to existing approaches the focus should be on two questions. First, does the platform offer a high index contrast? Many of the advantages of subwavelength structure arise from the ability to tune the refractive index over a wide range, and given the nonresonant nature of subwavelength structures, this is only possible within the limits of the available index contrast. Second, is it possible to fabricate small enough feature sizes to achieve subwavelength operation? In this sense, CMOS-compatible materials such as silicon nitride and germanium will likely be explored, and more exotic materials such as diamond will also become of interest.


Corresponding author: José Manuel Luque-González, Telecommunication Research Institute (TELMA), Universidad de Málaga, CEI Andalucía TECH, Louis Pasteur 35, 29010 Málaga, Spain, E-mail:

Funding source: National Research Council of Canada (NRC) Collaborative Science, Technology and Innovation Program (CSTIP)

Award Identifier / Grant number: HTSN 210

Award Identifier / Grant number: HTSN 209

Funding source: Universidad de Málaga

Funding source: Ministerio de Educación, Cultura y Deporte

Award Identifier / Grant number: FPU16/03401

Award Identifier / Grant number: FPU16/06762

Funding source: Ministerio de Ciencia Innovación y Universidades (MCIU)

Award Identifier / Grant number: PID2019-106747RB-I00

Funding source: Consejería de Economía, Conocimiento, Empresas y Universidad (CECEU)

Award Identifier / Grant number: P18-RT-793

Award Identifier / Grant number: UMA18-FEDERJA-219

Award Identifier / Grant number: P18-RT-1453

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The authors acknowledge funding from Universidad de Málaga, Ministerio de Ciencia, Innovación y Universidades (MCIU) (PID2019-106747RB-I00), Ministerio de Educación, Cultura y Deporte (MECD) (FPU16/06762, FPU16/03401); Consejería de Economía, Conocimiento, Empresas y Universidad (CECEU) (UMA18-FEDERJA-219, P18-RT-1453, P18-RT-793) and National Research Council of Canada (NRC) Collaborative Science, Technology and Innovation Program (CSTIP) (HTSN 209, HTSN 210).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Supplementary material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2021-0110).

Received: 2021-03-15
Accepted: 2021-07-19
Published Online: 2021-08-13

© 2021 José Manuel Luque-González et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.