Anisotropic metamaterials for scalable photonic integrated circuits: a review on subwavelength gratings for high-density integration

3.1 Cartesian coordinates

In conventional total internal reflection (TIR), when a propagating light in a dielectric is reflected at the interface of an infinite dielectric with a lower refractive index, light is totally reflected back if the incident angle is larger than a critical angle, which depends on the index ratio of the two mediums: θ _c = arcsin(n ₂/n ₁), where n ₁ and n ₂ are the refractive index of the first and the second mediums, respectively. When TIR happens, light evanescently decays in the second medium. The skin depth is the distance at which the field magnitude drops to 1/e. Skin depth defines the fundamental limit of light confinement in dielectric waveguides and resonators. The skin depth, δ, in isotropic media only depends on the index contrast and the incident angle:

where k ₀ is the wavevector in the vacuum and θ _in is the incident angle in the first medium, assuming it is greater than the critical angle (θ _in > θ _c ). The lower limit of the skin depth in isotropic dielectrics (n > 1) is δ = 1 / k 0 n 1 2 − 1 . Hence, to reduce the skin depth and confine light in the first medium, it is required to use a high-index dielectric as the first medium. However, there is an upper limit for the dielectric constant of transparent dielectrics at optical frequencies, which limits the control of evanescent fields.

When the second medium is anisotropic, another degree of freedom can also control the skin depth [75]:

where n _2x and n _2z are the refractive indices of the second medium parallel and perpendicular to the interface, respectively. It is also seen in Eq. (2) that the TIR condition is relaxed to:

We call this new condition relaxed-TIR [76] as it allows us to use the other component of the refractive index independently to control the skin depth. Thus, not only by increasing the contrast between n ₁ and n _2x , but also by increasing the anisotropy ratio (n _2z /n _2x ≫ 1) we can reduce the skin depth [Figure 2(b)] [22], [75]. It is also possible to control the degree of anisotropy in the other way (n _2z /n _2x ≪ 1) to extend the penetration of evanescent fields in the second medium [77]. It is worth noting that this skin-depth engineering effect has also been demonstrated in 2D materials exhibiting strong anisotropy [22], [78], [79].

3.2 Cylindrical and spherical coordinates

Skin-depth engineering of the TM modes in cylindrical coordinates, when the optical axis is in the z direction, is similar to that in the cartesian coordinates [73]. When an isotropic cylinder with a permittivity of ɛ ₁ is surrounded by an anisotropic cladding (ɛ _2ρ = ɛ _2φ = ɛ _2⊥ ≠ ɛ _2z ), the radial wave-vector in the cladding is:

where β is the wave-vector in the z direction (e.g. propagation constant in optical fibers). If k _2ρ becomes imaginary, the field becomes evanescent in the cladding, and by increasing ε 2 z / ε 2 ⊥ , the skin depth can be reduced [73].

When the optical axis is radial, the skin depth engineering in the cylindrical and spherical coordinates becomes more complicated. This is because the anisotropy does not impact the radial wavevector, but it impacts the angular momentum wave number.

In order to analyze this, we consider the wave equations in media exhibiting spherical anisotropy. A similar behavior is observed in cylindrical coordinates [50]. The wave equation for a uniaxial dielectric, with its optical axis aligned along the r direction (ɛ _θ = ɛ _φ = ɛ _⊥), can be expressed as [74]:

where ℏ L ⃗ = ℏ i ( r ⃗ × ∇ ⃗ ) is the angular momentum operator with an eigenvalue of ℏ n ( n + 1 ) and n is an integer number describing the angular momentum mode number. Note that our focus here is on TM modes, as TE modes remain unaffected by dielectric anisotropy. The first term on the left-hand side of Eq. (5) corresponds to the radial momentum with an eigenvalue of ℏk _r which can be expressed as [74]:

The expansion of a plane wave in spherical coordinates with radial anisotropy can be written as [74]:

where P n ( 1 ) is the associated Legendre polynomial of the first order, z _n is one of the Riccati–Bessel functions or their superposition, and c _n is the amplitude of the nth-mode.

By increasing n in Eq. (6), the radial momentum reduces, and when k 0 r < n ( n + 1 ) / ε r , it becomes imaginary. This causes the field to decay faster when it approaches toward the center. The modes with higher angular momentum vanish faster. Increasing ɛ _⊥/ɛ _r leads to a faster decay of evanescent fields. This type of anisotropic media can be utilized as a shell in a core–shell resonator to confine evanescent waves inside the dielectric core and to increase Q of the resonators [80], [81], [82].

On the other hand, if we increase the anisotropy in the opposite direction (ɛ _⊥/ɛ _r ≪ 1), near-field evanescent waves can be enhanced without a significant change in the momentum away from the center. The field enhancement using this approach in the subwavelength regime is more substantial than increasing the permittivity in isotropic media [74]. This can lead to a strong conversion of reactive (evanescent) fields near the center into propagating electromagnetic waves even without using metal [50].

A resonator composed of a subwavelength low-index dielectric surrounded by an anisotropic shell (ɛ _⊥/ɛ _r ≪ 1) can enhance the light intensity at the core/shell interface [Figure 2(c) and (d)] due to the enhancement of evanescent fields. The continuity of the displacement current at the boundary also helps to increase the field further without significantly increasing the Q [74], [83]. This method can be applied on a chip to demonstrate subwavelength nano-antennas with high scattering efficiency.

4.1 Anisotropic perturbation with eskid waveguide: TE zero-crosstalk

Figure 3(a) illustrates the schematics of the coupled strip and the eskid waveguides. The SWG claddings of the eskid waveguide can be represented as a homogenized metamaterial using the EMT shown in Figure 1. Since this EMT model effectively captures the fundamental impact of the SWG cladding on the coupled eskid waveguides, it will be used for the following analysis. The coupling strength between the waveguides can be quantified by the overall coupling coefficient |κ| = |κ _x + κ _y + κ _z |, where each component κ _i (i = x, y, and z) is calculated by [59], [97]:

Here, E _1i (x, y) and E _2i (x, y) represent the i-components of the normalized mode profiles for unperturbed waveguide modes 1 and 2, respectively, and Δɛ(x, y) denotes the perturbation index. To calculate Eq. (8), the index distribution ɛ(x, y) of the coupled eskid waveguides can be decomposed as shown in Figure 3(b): the static index ɛ _s(x, y) representing the background (i.e., except the core regions), and the perturbation indices Δɛ ₁(x, y) and Δɛ ₂(x, y) due to the presence of each waveguide. The normalized field profiles E ₁ (x, y) and E ₂ (x, y) used in Eq. (8) are calculated with the index profiles ɛ ₁(x,s y) = ɛ _s(x, y) + Δɛ ₁(x, y) and ɛ ₂(x, y) = ɛ _s(x, y) + Δɛ ₂(x, y), respectively. The key factor in determining the coupling coefficient is the field interaction within the perturbation region. For example, if we set the waveguide 2 (right-side) as the perturbation (i.e., Δɛ = Δɛ ₂), then the interaction between the evanescent field of E _1i and the confined field of E _2i determines κ _i . The sign of the coupling coefficient can be either positive or negative, depending on the field distributions of E _1i and E _2i in this perturbation region. By placing anisotropic SWG cladding [Figure 1(b)], the evanescent field can be engineered anisotropically, manipulating the coupling coefficient to achieve even κ = 0; this is the key idea behind anisotropic perturbation to achieve a zero-crosstalk.

Figure 3(c) and (d) respectively plot the real E _1x and the imaginary E _1z of the TE mode in eskid (solid) and strip (dashed), along the x-axis (at y = h/2). The contribution of E _1y is negligible. The inset illustrates the simulated index profile ɛ ₁(x, y). The yellow-shaded region highlights the perturbation region that determines the coupling coefficient. As expected, in Figure 3(c), the skin depth of E _1x is significantly reduced in the eskid configuration compared to the strip, resulting in a lower κ _x for the eskid waveguide. However, in Figure 3(d), the eskid has a smaller effect on E _1z compared to E _1x , resulting in a comparable |κ _z | as in strip. This anisotropic skin depth effect reduces the difference in magnitude between κ _x and κ _z , and since these coupling coefficients have opposite signs, the overall coupling coefficient |κ| can be minimized or even reduced to zero by tailoring the geometric parameters. The detailed results are shown in Figure 4(e).

Figure 4:

Dielectric perturbation and coupling length comparisons between waveguides with different anisotropic claddings for TE and TM [94], [95]. Dielectric perturbation coefficients are plotted to illustrate the coupling mechanisms. (a–c) Schematics of the coupled strip, eskid, and leaky-SWG waveguides, respectively. (d–i) Normalized coupling coefficients κ _x (purple dashed), κ _y (blue dashed), and κ _z (green dashed), along with the overall coupling coefficient |κ| = |κ _x + κ _y + κ _z | (red solid), are plotted as a function of SWG width w _swg. The corresponding normalized coupling length L _c/λ ₀ (solid blue line) is shown below in each plot. (d–i) Represent TE and TM polarizations, respectively, for each configuration: (d, g) strip, (e, h) eskid, and (f, i) leaky-SWG waveguides. The effective medium theory is used to model the ɛ _‖ and ɛ _⊥ in (b) and (c) schemes. Note that, for TE in the eskid (e), the overall |κ| approaches zero, and the coupling length increases to infinite, due to anisotropically suppressed field perturbations as shown in Figure 3(c) and (d). Similarly, for TM in the leaky-SWG (i), the |κ| becomes zero, resulting in an infinite coupling length, but due to anisotropic leaky-like oscillations as shown in Figure 3(g) and (h). Geometric parameters are height h = 220 nm, core width w ₀ = 530 nm, and g = 65 nm, with ρ = 0.45, and a wavelength of λ ₀ = 1,550 nm. Reproduced with permission from Ref. [94], [95].

4.2 Anisotropic perturbation with leaky-SWG: TM zero-crosstalk

The idea of skin depth suppression used in the eskid waveguide is effective only for TE. For TM, the eskid would rather increase the crosstalk. Unlike in TE, the coupling coefficients of a guided TM mode do not have the negative κ _i terms that are essential for anisotropic perturbation. To achieve the anisotropic perturbation in TM, a waveguide design with SWG claddings aligned along the propagation direction was introduced, as shown in Figure 3(e) [95]. In this scheme, minor coupling coefficients κ _x and κ _z emerge due to oscillating field behavior within the cladding region. This oscillation is caused by the radiative leaky mode, which generally leads to strong coupling between waveguides, even at a far distance, when the cladding materials are isotropic [98], [99]. However, with anisotropic SWGs, the oscillatory behavior of each field component can be controlled anisotropically. This control enables the minor coupling coefficients κ _x and κ _z to counteract and compensate for the dominant coupling coefficient κ _y , thereby reducing the overall coupling.

Figure 3(f) illustrates the conceptual formation of the anisotropic leaky-like SWG mode. Figure 3(g)–(i) plot the Re(E _1y ), Re(E _1x ), and Im(E _1z ) for both leaky-SWG (solid) and strip (dashed) configurations along the x axis (at y = h/2). Similar to the TE, the inset shows the simulated index profile ɛ ₁(x, y), with the yellow-shaded region indicating the perturbation region Δɛ ₂. For the dominant field Re(E _1y ), the difference between leaky-SWG and strip configurations is minimal [Figure 3(g)]. However, the magnitude of E _1x in the leaky-SWG significantly increases in the perturbation region compared to the strip, leading to a non-negligible κ _x , which counterbalances the κ _y [Figure 3(h)]. For the z-component, the sign of E _1z is reversed in the perturbation region, becoming opposite to that of E _2z , resulting in an overall negative κ _z [Figure 3(i)]. As a whole, the SWGs claddings induce negative κ _x and κ _z that counteract the positive κ _y , reducing the overall coupling coefficient even in the presence of a radiative leaky mode. By optimizing the geometric parameters, further tuning can achieve zero-|κ|, as in Figure 4(i).

4.3 Coupling length comparison

The coupling coefficients and the corresponding coupling length L _c /λ ₀ of the strip, eskid, and leaky-SWG waveguides are summarized and compared in Figure 4. Figure 4(a)–(c) illustrate the schematics of the coupled strip, eskid, and leaky-SWG waveguides, respectively. Figure 4(d)–(i) plot the simulated coupling coefficients and L _c /λ ₀ for TE and TM modes, respectively, as a function of SWG width w _swg. In these figures, the normalized κ _x , κ _y , and κ _z are represented by purple, blue, and green dashed lines (left axis), while the magnitude of the total coupling coefficient |κ| is shown as a red solid line (right axis). The corresponding coupling length is calculated by L _c = π/(2|κ|) and plotted below with a solid blue line. All SWGs are modeled using the EMTs.

For TE mode, where κ _x is dominant [Figure 4(d)–(f)], κ _y is negligible and κ _z is negative in all configurations. The negative κ _z reduces the coupling due to κ _x . In all configurations, the coupling coefficients decrease as w _swg increases due to the reduced field overlap, leading to longer coupling lengths. In a typical strip waveguide, κ _x is dominant, resulting in a finite coupling length [Figure 4(d)]. However, in the eskid, skin depth suppression significantly reduces |κ _x |, as shown by the coupling length plot. In general, the L _c /λ ₀ of the eskid [Figure 4(e)] is approximately one order of magnitude longer than that of the strip waveguide [Figure 4(d)]. Furthermore, at a specific SWG width (w _swg ≈ 470 nm), κ _z completely cancels out the κ _x , making the overall coupling coefficient zero (κ = 0). This corresponds to an infinitely long coupling length, i.e., L _c = ∞. In contrast, for the leaky-SWG configuration, the anisotropic cladding indices ɛ _x and ɛ _z are reversed, extending the evanescent wave and causing larger crosstalk [Figure 4(f)]. Although this configuration is not ideal for low crosstalk, it can be used for an efficient coupler with a shorter coupling length [73], [77].

For TM mode, where κ _y is dominant [Figure 4(g)–(i)], other κ _x and κ _z are relatively negligible in guided modes, as shown in Figure 4(g) and (h). Only the leaky-SWG configuration exhibits non-negligible κ _x and κ _z [Figure 4(i)], due to the oscillatory field pattern in the cladding associated with the leaky radiation. The κ _x and κ _z are negative due to the field overlaps shown in Figure 3(h) and (i). Unlike the other guided modes, the magnitudes of κ _x and κ _z increase as w _swg enlarges due to the oscillating field nature in the cladding. This allows for the zero total coupling coefficient |κ| = 0, leading to an infinitely long coupling length L _c = ∞. Here, this is achieved near w _swg ≈ 630 nm. On the other hand, the eskid generates positive κ _z , resulting in a higher |κ| and a shorter coupling length [Figure 4(h)].

4.4 Experimental demonstrations of zero-crosstalks in TE and TM

The zero-crosstalk responses in TE and TM modes, as discussed in Section 4.3, has been experimentally demonstrated. Figure 5(a)–(d) show the schematic and SEM images of the fabricated devices, with SWG arrays parallel (eskid) and perpendicular (leaky-SWG) to the propagation direction, respectively. The output power ratio I ₂/I ₁, which defines optical crosstalk, was measured by sending light through one of the coupled waveguides. The coupling length L _c was then extracted using the following relation [59]:

where L is the physical length of the coupled waveguides, and I ₁ and I ₂ are output powers at the through and coupled ports, respectively.

Figure 5:

Experimental demonstrations of zero-crosstalk singularity in the eskid and leaky-SWG waveguides. (a–c) Eskid waveguide with TE mode [94]; (d–f) leaky-SWG waveguide with TM mode [95]. Schematics and SEM images of the fabricated (a) eskid and (d) leaky-SWG waveguides. (b) Simulated and (c) experimentally characterized coupling lengths L _c/λ ₀ as functions of free-space wavelength λ ₀ for different waveguide widths: w = 420 nm (blue), w = 430 nm (red), w = 440 nm (yellow), and w = 450 nm (purple). Solid and dashed lines represent the eskid and strip waveguides, respectively. (e) Simulated and (f) experimentally characterized L _c/λ ₀ as functions of λ ₀ for different waveguide widths: w = 565 nm (red), w = 570 nm (blue), and w = 575 nm (green), with solid and dashed lines representing the leaky-SWG and strip waveguides, respectively. The peaks in L _c/λ ₀ represent zero-crosstalk singularities with infinitely long coupling lengths. Reproduced with permission from Refs. [94], [95].

For TE mode, Figure 5(b) and (c) show the L _c/λ spectra of coupled eskid (solid) and strip (dashed) waveguides: (b) simulations and (c) experiments. Each color represents a different core width, while fixing the w _swg. In conventional coupled strip waveguides (dashed lines), the coupling lengths are less than ≈ 1 0 3 wavelengths, regardless of core width w. However, in the coupled eskid waveguides (solid lines), L _c/λ is extended by one to two orders of magnitude, primarily due to the reduced skin-depth effect. In addition, depending on the core width, a coupling length singularity appears where L _c /λ = ∞, originating from the anisotropic perturbation effect. In the experiment, the peak L _c/λ observed is around 10⁵, which is likely limited by detector sensitivity or background noise of the system. Nevertheless, the significant increase in coupling length is clearly shown, demonstrating the longest coupling length.

Similarly, Figure 5(e) and (f) show the L _c/λ spectra for the coupled leaky-SWG waveguides with TM mode. Due to the lower confinement of TM compared to TE, the coupling lengths of the strip (dashed lines) are less than 10² wavelengths. However, in the leaky-SWG configuration, L _c/λ can be increased by approximately 10⁴ wavelengths, though the experimental results are again limited by the measurement system. Since the anisotropic perturbation is sensitive to geometric parameters, the corresponding spectrum also exhibits high sensitivity. This feature makes the leaky-SWG configuration particularly suitable for applications such as refractive index sensors [100]. Further optimization through dispersion engineering or tapering could help reduce the spectral sensitivity and broader bandwidth, providing more stable device performance.

5.1 Confinement manipulation

The fundamental basis of a PIC is the waveguide mode, and manipulating its confinement by engineering the core/cladding indices (and thus the index contrast) and their anisotropy plays a crucial role in overall device performance. Each component may require either high or low confinement, depending on its specific functionality. Here, the SWGs, with their index modulation effect and anisotropic characteristics as discussed in previous sections, significantly expand the range of confinement control. This flexibility has made SWGs a key tool for tailoring confinement engineering.

For example, the zero-crosstalk response introduced in Section 4 can be directly employed for a densely packed delay line [Figure 6(a)] [101] and a highly selective polarization beam splitter (PBS) [Figure 6(b)] [102]. In all these cases, the tight field confinement enabled by the anisotropic nature of SWGs is leveraged to enhance device performance. This tight confinement also helps reduce bending loss by suppressing radiation losses, making it effective even in high-radiative-loss regimes where small bending radii are required [93], [124]. This enables the design of a compact TE pass polarizer, where only the TE mode is highly confined with reduced bending loss [Figure 6(c)] [103]. When integrated into both the core and cladding, SWGs allow greater flexibility in adjusting index contrast and anisotropy, thereby improving confinement control [125]. A waveguide crossing bar is another good example. In Figure 6(d) [47], 1D and 2D SWGs are combined to create a compact crossing bar while effectively suppressing unwanted side couplings. The 1D SWG acts as an anisotropic cladding, reducing the skin depth to prevent coupling to unintended ports, while the 2D SWGs facilitate smooth modal transitions, minimizing reflections from the 1D SWG regions. Alternatively, more complex SWG designs, such as nanohole SWGs and fan-shaped bent SWGs, can be used to manipulate the modal phase and index distribution, further enhancing the focusing of multimodal interference crossing bars [Figure 6(e)] [104].

Conversely, reducing the index contrast between the core and cladding allows light to leak into the cladding, extending the evanescent field and making it easier to couple light into other structures. These characteristics allow for rapid light transitions to desired locations. For example, in Figure 6(f) [105], the TM-pass splitter functions as a Bragg reflector for TE while acting as a directional coupler for TM. The SWGs in this design are optimized to create strong evanescent coupling for TM, enabling fast power transitions. Similarly, in the optical phased array shown in Figure 6(g) [106], the core index is reduced using SWGs, which enhances coupling to adjacent emitters and improves radiation efficiency for beam steering.

SWGs can also modulate modal field distributions and exploit optical properties, beyond merely adjusting light confinement. For example, by fabricating trapezoidal SWGs with different duty cycles, light can be biased towards the larger duty cycle, resulting in an asymmetric light distribution. This can be used to achieve faster light coupling in the power splitter [126] or to concentrate light within the inner side of a ring resonator, reducing bending loss and improving the quality factor [Figure 6(h)] [107]. Additionally, apodizing the filling fraction of SWGs can modulate the diffraction levels, creating a Gaussian mode distribution that enhances coupling efficiency [Figure 6(i)] [108]. SWGs can also be used to form and modulate a slot mode, which strongly confines light in the middle of a low-index region [Figure 6(j)] [109], [127].

5.2 Hetero-anisotropy and zero-birefringence

One of the most distinctive properties that SWGs offer is their ability to control anisotropy. The anisotropy in a structure can either be uniformly distributed or vary across different regions, which is referred to as hetero-anisotropy. The anisotropy of SWGs can be tailored by adjusting the filling fractions, orientations, and shapes of the gratings, making them ideal for implementing and manipulating hetero-anisotropy in PICs. SWGs can manipulate TE and TM modes differently within the same structure, a feature that is effectively applied in polarization handling devices such as PBSs and polarization splitters and rotators (PSRs).

Extensive research has led to the development of advanced SWG-supported PBSs that leverage hetero-anisotropy for enhanced performance. For example, the PBS shown in Figure 6(k) confines the TE to the upper waveguide due to the reduced skin depth caused by anisotropy, while the TM passes to the lower waveguide through a multimode interference (MMI) [110]. The inclusion of anisotropic SWGs can broaden the MMI bandwidth [42], [128] and further optimization is achieved by tilting the SWGs [129]. Similarly, in Figure 6(l), TM propagates through a homogeneous material to the lower port without experiencing total internal reflection (TIR), while TE undergoes TIR and is directed to the upper port [111]. This design, based on the Glan–Thompson prism, uses anisotropic cement to reduce the skin depth and enhance TE confinement, improving polarization selectivity. In Figure 6(m), two orthogonal SWG core waveguides are coupled, forming an efficient directional coupler for the TM and a low-crosstalk waveguide for the TE, thus effectively enabling polarization splitting [112].

While hetero-anisotropy is often used to enhance anisotropic behavior, SWGs can also be engineered to reduce modal anisotropy and achieve zero-birefringence. Generally, in PICs, the waveguide width is larger than the height for easier etching during fabrication. This leads to a higher effective index for TE than TM, resulting in significant birefringence and polarization sensitivity. However, this issue can be addressed by carefully engineering the anisotropy of SWGs, making it possible to equalize the modal indices for both TE and TM modes. For example, when SWGs are placed parallel to the propagation direction [as in Figure 1(b)], the permittivity in the TE direction (ɛ _x = ɛ _⊥) is lower than that in the TM direction (ɛ _y = ɛ _‖), since ɛ _⊥ < ɛ _‖. Therefore, by tuning the duty cycle, it is possible to equalize the refractive indices ɛ _x and ɛ _y equal, even in structures with asymmetric height-to-width aspect ratio. Similarly, zero-birefringence can also be achieved using 2D SWG patterns [Figure 6(n)]. This approach was adopted for the development of polarization-independent grating couplers [48], [113]. Moreover, alternatively, tilting the SWGs at a specific angle can achieve a zero-birefringence waveguide, as in Figure 6(o) [114]. While the TM mode’s modal index remains largely unaffected by SWG tilt, the TE mode’s index varies significantly with the tilt angle, enabling TE and TM index matching through angle optimization.

5.3 Adiabatic mode conversion

In many PIC components, adiabatic mode conversion is widely used to minimize losses caused by abrupt modal transitions. This can be achieved through gradual structural changes, which prevent discontinuities and maintain a continuous state. Such smooth transition between modes also allows for a broader bandwidth, improving fabrication tolerance. A simple method for adiabatic mode conversion is geometric tapering, where the index distribution is adjusted gradually along the propagation direction, enabling efficient mode transition with high conversion efficiency. However, the gradual transition often leads to a larger footprint, limiting the overall compactness of devices. Here, SWGs provide extra degrees of freedom for efficient modal transitions, for example, by allowing index redistribution through adjustments in the filling fraction. SWGs are often combined with geometric tapering to enhance modal conversion efficiency and minimize device size [130], [131], [132], [133]. Tuning the duty cycle of SWGs also can lower the index, simplifying the design and making it more tolerant to fabrication errors.

A widely explored application of this approach is a fiber-to-chip edge coupler [24], [25], [26], [52], [53], [54]. Edge couplers generally offer broader bandwidth than grating couplers but have larger footprints due to the tapering required for mode transition and are sensitive to fiber alignment with small mode field diameter (MFD) [134], [135]. By using SWGs as a core and adjusting their filling fraction, faster adiabatic transition can be achieved, resulting in more compact designs [Figure 6(p)] [115]. Moreover, using SWGs in the cladding with a lower filling fraction can reduce the index contrast, thereby increasing the MFD and improving alignment tolerance [54]. Numerous innovative SWG-based edge couplers have been explored extensively [52], [53], [136], [137].

Another key advantage of SWGs is their flexibility in shape and orientation, allowing for the creation of metamaterials with diverse properties beyond simple perpendicular or parallel arrangement. For example, gradient-index (GRIN) metalenses can be easily implemented by spatially varying the filling fractions of SWGs in different shapes. GRIN structures are particularly effective at expanding the mode size, making them ideal for applications such as spot size converter [Figure 6(q)] [55] and slab mode converter [Figure 6(r)] [56], achieving large modal conversion within a compact footprint. Various SWG shapes have been developed, including bricked SWGs for efficient mode-order transitions from fundamental TE to higher-order TE₁ and TE₂ modes [Figure 6(s)] [116]; polygonal SWGs for simultaneous phase and amplitude control to enable rapid mode conversion [Figure 6(t)] [117]; and N-shape tapered SWGs for a TM-pass polarizer, which enables selective TM mode transition while reflecting the TE mode in the Bragg regime [Figure 6(u)] [118].

Although only a few applications are introduced here, adiabatic mode conversion can be integrated into nearly all PIC components, often combined with other functions. Notable examples include adiabatic PBS [Figure 6(v)] [119] and PSR [Figure 6(w)] [120], where the anisotropic properties of SWGs, combined with adiabatic tapering, significantly enhance device performance. By leveraging these adiabatic mode transitions, SWGs enable efficient mode transitions, broader bandwidth, and improved fabrication tolerance.

5.4 Group velocity and dispersion control

Group velocity (v _g ) represents the speed at which the energy or information within the envelope of a wave propagates through a medium. Since group velocity strongly depends on modal confinement and wavelength dependency of effective indices, it can be precisely fine-tuned using the index engineering of SWGs. This tuning is essential for optimizing delay lines, managing dispersion and pulse propagation, and enhancing nonlinear optical processes.

For example, delay times in a delay line can be controlled by varying the filling fractions of SWGs rather than adjusting physical waveguide lengths. Traditional delay lines achieve longer delay times by increasing waveguide length, which typically leads to a larger footprint and increased signal loss. SWG-based delay lines, however, enable group velocity adjustment without changing length, resulting in more compact designs [Figure 6(x)] [121].

Index engineering of SWGs also provides control over modal dispersion [122]. Generally, waveguide modes with higher confinement exhibit anomalous dispersion, whereas lower confinement results in normal dispersion. By using SWGs to adjust modal confinement, a broad range of dispersion profiles can be synthesized [Figure 6(y)], which can be applied to optical communications, dispersion compensation, and optical signal processing.

Additionally, manipulating group velocity and dispersion through SWGs can significantly enhance nonlinear optical processes. One prominent example is four-wave mixing, a nonlinear interaction in which two or more waves combine to produce new optical frequencies. Achieving broadband phase matching in this process is challenging, but engineering SWGs to create a graded index profile allows different spatial modes with equidistant frequencies and a common wavevector, thus enabling broadband phase matching [138]. Another compelling nonlinear application is the generation of Bragg solitons through bandgap engineering of SWGs [123]. Near the band edge, strong anomalous dispersion can induce soliton compression, allowing on-chip Bragg soliton generation through precise tuning of the SWG grating pitch [Figure 6(z)]. Incorporating SWGs into the air cladding also allows for dispersion engineering that supports dispersive waveguide in supercontinuum generation [139].