Review on Optical Waveguides

Abstract

Optical devices are necessary to meet the anticipated future requirements for ultrafast and ultrahigh bandwidth communication and computing. All optical information processing can overcome optoelectronic conversions that limit both the speed and bandwidth and are also power consuming. The building block of an optical device/circuit is the optical waveguide, which enables low-loss light propagation and is thereby used to connect components and devices. This chapter reviews optical waveguides and their classification on the basis of geometry (Non-Planar (Slab/Optical Fiber)/Planar (Buried Channel, Strip-Loaded, Wire, Rib, Diffused, Slot, etc.)), refractive index (Step/Gradient Index), mode propagation (Single/Multimode), and material platform (Glass/Polymer/Semiconductor, etc.). A comparative analysis of waveguides realized in different material platforms along with the propagation loss is also presented.

1. Introduction

Waveguides are indispensable for communication and computing applications as they are immune to electromagnetic interference and induced cross talk and also counter diffraction. Next-generation high-end information processing (bandwidths >1 Tb/s and speed >10 Gb/s) is immensely challenging using copper-based interconnects. Optical interconnects transmit data through an optical waveguide and offer a potential solution to improve the data transmission [1, 2]. There are predominantly two classes of optical waveguide: those in which “classical optical elements, placed periodically along the direction of propagation of the wave, serve to confine the wave by successive refocusing in the vicinity of the optical axis (laser resonators and multiple lens waveguides); and those in which the guiding mechanism is that of multiple total internal reflection from interfaces parallel to the optical axis” (fiber optical waveguides, slab waveguides, and resonators) [3].

Historically, high-frequency microwave sources had created a furore on guided wave photonics pioneered by Rayleigh and Sommerfeld. The first theoretical description of mode propagation along a dielectric guide was done by Hondros and Debye in 1910 [3]. The first dielectric waveguide to be examined at optical frequencies was the glass fiber used primarily for fiber optics imaging applications [4].

A waveguide can be defined as any structure (usually cylindrical) used for guiding the flow of electromagnetic wave in a direction parallel to its axis, confining it to a region either within or adjacent to its surfaces. In order to understand the propagation of light in a waveguide, it is imperative to derive the wave equation. The electromagnetic wave equation can be derived from the Maxwell’s equation, assuming that we are operating in a source free $\left(\rho =0,J=0\right)$, linear $(\epsilon \text{and}\mu$ are independent of $E\text{and}H)$, and an isotropic medium. $E\text{and}H$ are the electric and magnetic field amplitudes, respectively, $\epsilon$ is the electric permittivity of the medium, and $\mu$ is the magnetic permeability of the medium. The equations are:

$$\nabla \times \overline{E}=-\frac{\partial \overline{B}}{\partial t}$$E1

$$\nabla \times \overline{H}=\frac{\partial \overline{D}}{\partial t}$$E2

$$\nabla .\overline{D}=0$$E3

$$\nabla .\overline{B}=0$$E4

Here, $B$ and $D$ are magnetic and electric fluxes, respectively. The wave equation derived from the above expressions is:

$${\nabla }^2\overline{E}-\mathit{\mu \epsilon }\frac{{\partial }^2\overline{E}}{\partial t^2}=-\nabla \left(\overline{E}.\frac{\nabla \epsilon }{\epsilon }\right)$$E5

The right-hand side of Eq. (5) is nonzero when there is a gradient in permittivity of the medium. Guided wave medium has a graded permittivity; however, in most structures, the term is negligible. Thus, the wave equation can be written as:

$${\nabla }^2\overline{E}-\mathit{\mu \epsilon }\frac{{\partial }^2\overline{E}}{\partial t^2}=0,{\nabla }^2\overline{H}-\mathit{\mu \epsilon }\frac{{\partial }^2\overline{H}}{\partial t^2}=0$$E6

for electric and magnetic field amplitudes, respectively.

2. Classification of waveguides

Optical waveguides can be classified according to their geometry, mode structure, refractive index (RI) distribution, and material. A dielectric optical waveguide comprises a longitudinally extended high-index medium called the Core, which is transversely surrounded by a low-index medium, called the Cladding. A guided optical wave propagates in the waveguide along the longitudinal direction. The characteristics of a waveguide are determined by the transverse profile of its dielectric constant (x, y), which is independent of the z coordinate. For a waveguide made of optically isotropic media, the waveguide can be characterized merely with a single spatially dependent transverse profile of the index of refraction, n(x, y). Broadly, the waveguides can be classified as [5]:

• Planar/2-D waveguides: Optical confinement is only in one transverse direction, the core is sandwiched between cladding layers in only one direction (Figure 1(a)). Optical confinement is only in the x-direction with index profile n(x). They are primarily used for high-power waveguide lasers and amplifiers.

• Non-planar/3-D/channel optical waveguide: Comprises of two-dimensional transverse optical confinement, the core is surrounded by cladding in all transverse directions, and n(x, y) is a function of both x and y coordinates as shown in Figure 1(b). A channel waveguide (with guidance in both directions) has a guiding structure in the form of a stripe with a finite width. Examples: channel waveguides (Section 2.3.II) and circular optical fibers [6].

Figure 1.

(a) Planar optical waveguide of 1-d transverse (x) optical confinement, (b) non-planar optical waveguide of 2-D transverse (x, y) optical confinement.

A waveguide in which the index profile changes abruptly between the core and the cladding is called a step-index waveguide, while one in which the index profile varies gradually is called a graded-index waveguide as shown in Figure 2. Recently, hybrid index profile waveguide was shown combining both inverse-step index waveguide and graded index waveguides for high-power amplification of a Gaussian single-mode beam [7].

Figure 2.

(a) Step-index type waveguide, (b) Graded-index waveguide, and (c) Hybrid waveguide.

2.1. Waveguide mode

A waveguide mode is an electromagnetic wave that propagates along a waveguide with a distinct phase velocity, group velocity, cross-sectional intensity distribution, and polarization. Each component of its electric and magnetic field is of the form $f\left(x,y\right)e^{\mathit{i\omega t}-\mathit{ihz}}$, where z is the axis of the waveguide. Modes are referred to as the “characteristic waves” of the structures because their field vector satisfies the homogenous wave equation in all the media that make up the guide, as well as the boundary conditions at the interfaces. The electric and magnetic fields of a mode can be written as $E_v\left(r,t\right)=E_v\left(x,y\right)\text{exp}\left(i{\beta }_{v}z-\mathit{i\omega t}\right)$ and $H_v\left(r,t\right)=H_v\left(x,y\right)\text{exp}\left(i{\beta }_{v}z-\mathit{i\omega t}\right)$, where ν is the mode index, $E_v\left(x,y\right)$ and $H_v\left(x,y\right)$ are the mode field profiles, and ${\beta }_v$ is the propagation constant of the mode.

A mode is characterized by an invariant transversal intensity profile and an effective index $(n_{\mathit{eff}}$). Each mode propagates through the waveguide with a phase velocity of $c/n_{\mathit{eff}}$, where c denotes the speed of light in vacuum and $n_{\mathit{eff}}$ is the effective refractive index of that mode. It signifies how strongly the optical power is confined to the waveguide core. In order to understand modes intuitively, consider a simple step-index 2-D waveguide and an incident coherent light at an angle θ between the wave normal and the normal to the interface as shown in Figure 3. The critical angle at the upper interface is ${\mathrm{\theta }}_c={\text{sin}}^{-1}{n}_c/n_f$ and lower interface ${\mathrm{\theta }}_s={\text{sin}}^{-1}{n}_s/n_f$ and ${\mathrm{n}}_s<{\mathrm{n}}_c({\mathrm{\theta }}_s<{\mathrm{\theta }}_{c)}$.

Figure 3.

Ray-optical picture of modes propagating in an optical waveguide.

Optical modes with an effective index higher than the largest cladding index are (1) Guided modes (${\mathrm{\theta }}_s<\mathrm{\theta }<{90}^{°})$: As the wave is reflected back and forth between the two interfaces, it interferes with itself. A guided mode can exist only when a transverse resonance condition is satisfied so that the repeatedly reflected wave has constructive interference with itself. Modes with lower index are radiating and the optical power will leak to the cladding regions. They can be categorized as (2) Substrate radiation modes (${\mathrm{\theta }}_c<\mathrm{\theta }<{\mathrm{\theta }}_s$): Total reflection occurs only at the upper interface resulting in refraction of the incident wave at the lower interface from either the core or the substrate, (3) Substrate-cover radiation modes ($\mathrm{\theta }<{\mathrm{\theta }}_c)$: No total reflection at either interface. Incident wave is refracted at both interfaces, and it can transversely extend to infinity on both sides of the waveguide, and (4) Evanescent modes: Their fields decay exponentially along the z direction. For a lossless waveguide, the energy of an evanescent mode radiates away from the waveguide transversely.

The waveguide dimensions determine which modes can exist. Most waveguides support modes of two independent polarizations, with either the dominant magnetic (quasi-TM) or electric (quasi-TE) field component along the transverse (horizontal) direction. For most applications, it is preferable that the waveguides operate in a single-mode regime for each polarization. This single-mode regime is obtained by reducing the waveguide dimensions until all but the fundamental waveguide modes become radiating. Fields in the waveguide can be classified based on the characteristics of the longitudinal field components, namely (1) Transverse electric and magnetic mode (TEM mode): $E_z=0$, and $H_z=0$. Dielectric waveguides do not support TEM modes, (2) Transverse electric mode (TE mode): $E_z=0$ and $H_z\ne 0$, (3) Transverse magnetic mode (TM mode): $H_z=0$ and $E_z\ne 0$, and (4) Hybrid mode: $E_z\ne 0$ and $H_z\ne 0$. Hybrid modes exist only in non-planar waveguide.

2.2. Planar waveguide

Homogeneous wave equations exist for planar slab waveguides of any index profile n(x). For a planar waveguide, the modes are either TE or TM.

Infinite slab waveguide: The slab waveguide is a step-index waveguide, comprising a high-index dielectric layer surrounded on either side by lower-index material (Figure 4). The slab is infinite in the y-z plane and finite in x direction and the refractive index of $n_{\mathit{\text{core}}}$ > $n_{\mathit{\text{cladding}}},n_{\mathit{\text{substrate}}}$ to ensure total internal reflection at the interface. For case (1): $n_{\mathit{\text{cladding}}}=n_{\mathit{\text{substrate}}}$, the waveguide is denoted as Symmetric and for case (2): $n_{\mathit{\text{cladding}}}\ne n_{\mathit{\text{substrate}}}$, waveguide is Asymmetric.

Figure 4.

Planar slab waveguide and transverse electric (TE) and transverse magnetic configuration (TM).

For the electromagnetic analysis of the planar slab waveguide (infinite width), assuming $n_{\mathit{\text{core}}}>n_{\mathit{\text{substrate}}}>n_{\mathit{\text{cladding}}}$, we consider two possible electric field polarizations—TE or TM. The axis of waveguide is oriented in z-direction: k vector of the guided wave will propagate down the z-axis, striking the interfaces and angles greater than critical angle. The field could be TE which has no longitudinal component along z-axis (electric field is transverse to the plane of incidence established by the normal to the interface, and the k vector) or TM depending on the orientation of the electric field.

I. For TE Asymmetric waveguide: E field is polarized along the y-axis, and assuming that waveguide is excited by a source with frequency ${\omega }_o$ and a vacuum wave vector of magnitude $\frac{{\omega }_o}{c}$, the allowed modes can be evaluated by solving the wave equation in each dielectric region through boundary conditions. For a sinusoidal wave with angular frequency ${\omega }_o$, the wave equation for the electric field components in each region can be written as ($\left|k\right|=\omega \sqrt{\mathit{\mu \epsilon }}=k$),

$${\nabla }^2{E}_y+{k_0}^2{n_i}^2{E}_y=0$$E7

here, $n_i$ can be the refractive index of either core, cladding, or the substrate. The solution to Equation (7) can be written as:

$$E_y\left(x,z\right)=E_y\left(x\right)e^{-\mathit{j\beta z}}$$E8

due to the translational invariance of the waveguide in z-direction. $\beta$ is the propagation constant along the z-direction (longitudinal). From Equation (8) and since $\frac{d^{2}y}{{\mathit{dx}}^2}=0$, we can write:

$$\frac{{\partial }^2{E}_y}{{\partial x}^2}+\left({k_0}^2{n_i}^2-{\beta }^2\right)E_y=0$$E9

The solution to the wave equation can be deduced by considering Case (1)$\beta >k_0{n}_i$ and $E_0$ is field amplitude at x = 0, solution is exponentially decaying and can be written as:

$$E_y\left(x\right)=E_0{e}^{\pm \sqrt{{\beta }^2-{k_0}^2{n_i}^2}x}$$E10

The attenuation constant $ϓ=\sqrt{{\beta }^2-{k_0}^2{n_i}^2}$. Case (2)$\beta <k_0{n}_i$, solution has an oscillatory nature and is given by:

$$E_y\left(x\right)=E_0{e}^{\pm \sqrt{{k_0}^2{n_i}^2-{\beta }^2}x}$$E11

The transverse wave vector $\mathrm{\kappa }=\sqrt{{k_0}^2{n_i}^2-{\beta }^2}$ and the relation between β, $\mathrm{\kappa }$ and k are given by $k^2={\beta }^2+{\kappa }^2$.

The longitudinal wave vector β (z component of k) must satisfy $k_0{n}_{\mathit{\text{substrate}}}<\beta <k_0{n}_{\mathit{\text{core}}}$ ($n_{\mathit{\text{cladding}}}\le n_{\mathit{\text{core}}}$) in order to be guided inside the waveguide. Eigen values for the waveguide can be derived using transverse components of electric field amplitudes in three regions as $E_y\left(x\right)=A{e}^{-{\gamma }_{\mathit{\text{cladding}}}x}\text{for}0<x;E_y\left(x\right)=\mathit{\text{Bcos}}\left({\kappa }_{\mathit{\text{core}}}x\right)+\mathit{\text{Csin}}\left({\kappa }_{\mathit{\text{core}}}x\right)\text{for}-h<x<0$; $E_y\left(x\right)=D{e}^{{\gamma }_{\mathit{\text{substrate}}}\left(x+h\right)}\text{for}x<-h,$ where A, B, C, and D are amplitude coefficients to be derived using boundary conditions ($E_y,H_z,\frac{\partial E_y}{\partial x}\left(\mathit{at}x=0\right)$ are continuous). Solving the equation, we get $A=B,C=-\frac{A{ϓ}_{\mathit{\text{cladding}}}}{{\kappa }_{\mathit{\text{core}}}}$ ,$D=A[\text{cos}\left({\kappa }_{\mathit{\text{core}}}h\right)+\frac{{\gamma }_{\mathit{\text{cladding}}}}{k_{\mathit{\text{core}}}}\text{sin}(k_{\mathit{\text{core}}}h]$, and thus:

$$\begin{array}{cc}{\left.\frac{\partial E_y}{\partial x}\right|}_{x=-h}& =A\left[k_{\mathit{\text{core}}}\text{sin}\left(k_{\mathit{\text{core}}}h\right)-{\gamma }_{\mathit{\text{cladding}}}\mathit{cos}\left(k_{\mathit{\text{core}}}h\right)\right]\text{(Core)}\\ & =A\left[\mathit{cos}\left(k_{\mathit{\text{core}}}h\right)+\frac{{\gamma }_{\mathit{\text{cladding}}}}{k_{\mathit{\text{core}}}}\text{sin}\left(k_{\mathit{\text{core}}}h\right)\right]{\gamma }_{\mathit{\text{substrate}}}\text{(Substrate)}.\end{array}$$

The Eigenvalue equation (Figure 5(a)) is given by:

$$\text{tan}\left(h{k}_{\mathit{\text{core}}}\right)=\frac{{\gamma }_{\mathit{\text{cladding}}}+{\gamma }_{\mathit{\text{subtrate}}}}{k_{\mathit{\text{core}}}\left[1-\frac{{\gamma }_{\mathit{\text{cladding}}}{\gamma }_{\mathit{\text{substrate}}}}{k_{\mathit{\text{core}}}^2}\right]}$$E12

Figure 5.

Plot for Eigen value equation for (a) Asymmetric TE mode slab waveguide, (b) Asymmetric TM mode slab Waveguide.

II. TM Asymmetric waveguide: The field components of the waveguide can be written as:

$H_Y\left(x,y,t\right)=H_m\left(x\right)e^{-\mathit{i\omega t}}$, $E_x\left(x,z,t\right)=\frac{\beta }{\mathit{\omega \epsilon }}{H}_m\left(x\right)e^{-i\left(\mathit{\omega t}-\mathit{\beta z}\right)}$, and $E_z\left(x,z,t\right)=\frac{-i}{\mathit{\omega \epsilon }}.$ The Eigen value for β (Figure 5(b)) is given by:

$$\text{tan}\left(h{k}_{\mathit{\text{core}}}\right)=\frac{k_{\mathit{\text{core}}}\left[\frac{n_{\mathit{\text{core}}}^2}{n_{\mathit{\text{substrate}}}^2}{\gamma }_{\mathit{\text{substrate}}}+\frac{n_{\mathit{\text{core}}}^2}{n_{\mathit{\text{cladding}}}^2}{\gamma }_{\mathit{\text{cladding}}}\right]}{k_{\mathit{\text{core}}}^2-\frac{n_{\mathit{\text{core}}}^4}{n_{\mathit{\text{cladding}}}^2{n}_{\mathit{\text{substrate}}}^2}{\gamma }_{\mathit{\text{cladding}}}{\gamma }_{\mathit{\text{substrate}}}}$$E13

III. TE Symmetric waveguide: The field equation of a TE mode within the symmetric waveguide is given by:

$$\begin{array}{ll}{E}_y=A{e}^{-\gamma \left(x-\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}& \text{for} \mathrm{x}\ge \raisebox{1ex}{$\mathrm{h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ E_y=A\frac{\text{cos}\mathit{\kappa x}}{\text{cos}\raisebox{1ex}{$\mathit{\kappa h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathit{\text{or}}A\frac{\text{sin}\mathit{\kappa x}}{\text{sin}\raisebox{1ex}{$\mathit{\kappa h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}& \text{for}-\raisebox{1ex}{$\mathrm{h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \mathrm{x}\le \raisebox{1ex}{$\mathrm{h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ E_y=\pm A{e}^{\gamma \left(x+\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}& \text{for}\mathrm{x}\le -\raisebox{1ex}{$\mathrm{h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}$$E14

The characteristic Eigen value equation for the TE modes in a symmetric waveguide is given by:

$$\begin{array}{ll}\text{tan}\frac{\mathit{\kappa h}}{2}& =\raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$k$}\right. \text{for even (cos) modes}\\ & =-\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$y$}\right. \text{for odd (sin) modes}\end{array}$$E15

In order to plot the Eigen values of the TE modes of the symmetric waveguide, solutions of Eq. (15) are plotted for a wavelength of 1.55 μm and different “h” values (15 μm and 3 μm respectively) as shown in Figure 6.

Figure 6.

Plot for Eigen value equation for Symmetric TE mode slab waveguide at a wavelength of 1.55 μm for waveguide width of (a) 15 μm and (b) 3 μm. Thick waveguide supports multimode transmission.

The longitudinal wave vector β is quintessential to describe the field amplitudes in all regions of the waveguide. (i) Every Eigen value β corresponds to a distinct confined mode of the system. The amplitude of the mode is established by the power carried in the mode; (ii) only a finite number of modes will be guided depending on the wavelength, index contrast, and waveguide dimensions; (iii) most modes will be unguided, and all modes are orthogonal to each other; (iv) some modes are degenerate. Degenerate modes will share the same value of β but will have distinguishable electric field distributions. The lower-order mode is expressed by ${\beta }_{\mathit{\text{lowest order}}}\approx k{n}_{\mathit{\text{core}}}$ and higher-order mode by ${\beta }_{\mathit{\text{lowest order}}}\approx k{n}_{\mathit{\text{core}}}\mathit{cos}{\theta }_{\mathit{\text{critical}}}\approx k{n}_{\mathit{\text{substrate}}}$. A waveguide is generally characterized by its normalized frequency, given by,

$V=\mathit{hk}{\left(n_{\mathit{\text{core}}}^2-n_{\mathit{\text{subtrate}}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$.

The approximate number of modes (m) in the waveguide are given by $m\approx V/\pi$. Graphical solution to the waveguide can be evaluated by:

$$V=k_{0}h{\left(n_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$E16

$$a=\raisebox{1ex}{$(n_{\mathit{\text{substrate}}}^2-n_{\mathit{\text{claddding}}}^2)$}\!\left/ \!\raisebox{-1ex}{$\left(n_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}}^2\right)$}\right.$$E17

$$b=\raisebox{1ex}{$(n_{\mathit{eff}}^2-n_{\mathit{\text{substrate}}}^2)$}\!\left/ \!\raisebox{-1ex}{$\left(n_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}}^2\right)$}\right.$$E18

where a is asymmetry parameter (ranges from 0 (symmetric waveguide) to infinity), b is normalized effective index (ranges from 0 (cutoff) to 1) and $n_{\mathit{eff}}=\beta /k_o$ is the effective index of the waveguide. The normalized dispersion relation is given by (Figure 7):

$$V\sqrt{1-b}=v\mathrm{\pi }+{\text{tan}}^{-1}\sqrt{b/\left(1-b\right)}+{\text{tan}}^{-1}\sqrt{\left(b+a\right)/\left(1-b\right)}$$E19

where ν is an integer. The cut-off condition (b = 0) for modes in a step-index waveguide is given by $V={\mathit{tan}}^{-1}\sqrt{a}+v\mathrm{\pi }$. The numerical aperture is defined as the maximum angle that an incident wave can have and still be guided within the waveguide. It is given by: N $A=\mathit{sin}{\theta }_{\mathit{max}}=\sqrt{n_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}/\mathit{\text{cladding}}}^2}$.

Figure 7.

Normalized index b versus normalized frequency V for different values of asymmetry coefficient a (a = 0, a = 10, a = $\infty$).

IV. TM Symmetric waveguide: The characteristic Eigen value equation for the TM modes in a symmetric waveguide is given by:

$$\begin{array}{ll}\text{tan}\frac{\mathit{\kappa h}}{2}& ={\left(n_{\mathit{\text{core}}}/n_{\mathit{\text{substrate}}}\right)}^2\raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$\kappa $}\right.: \text{even (cos) modes}\\ & =-{\left(n_{\mathit{\text{core}}}/n_{\mathit{\text{substrate}}}\right)}^2\raisebox{1ex}{$\kappa $}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right. \text{for odd (sin) modes}\end{array}$$E20

Graphical solution to the waveguide can be evaluated using:

$$V=k_{0}h{\left(n_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$E21

$$a=\frac{n_{\mathit{\text{core}}}^2{n}_{\mathit{\text{substrate}}}^2-n_{\mathit{\text{cladding}}}^2}{n_{\mathit{\text{cladding}}}^2{n}_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}}^2}$$E22

$$b=\raisebox{1ex}{$(n_{\mathit{eff}}^2-n_{\mathit{\text{substrate}}}^2)$}\!\left/ \!\raisebox{-1ex}{$\left(n_{\mathit{\text{core}}}^2-n_{\mathit{\text{substrate}}}^2\right)$}\right.$$E23

2.3. Non-planar waveguide

The following section describes step-index circular and channel waveguides.

I. Step-index circular waveguide: The wave equation for the step-index circular waveguides in cylindrical coordinates is given by:

$$E\left(r,\varphi ,z\right)=\widehat{r}{E}_r\left(r,\varphi ,z\right)+\widehat{\varphi }{E}_{\varphi }\left(r,\varphi ,z\right)+\widehat{z}{E}_z\left(r,\varphi ,z\right)$$E24

At z = 0, field is purely radial (Figure 8).The $E_z$ component of the electric field couples only to itself and the scalar wave equation for $E_z$ is given by:

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial E_z}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{E}_z}{{\partial \varphi }^2}+\frac{{\partial }^2{E}_z}{{\partial z}^2}+k_0^2{n}^2{E}_z=0$$E25

Figure 8.

Schematic representation of step-index circular waveguide.

One can write $E_z\left(r,\varphi ,z\right)=R\left(r\right)\varphi \left(\phi \right)Z\left(z\right),$Eq. (24) can be written as:

$$R^{\prime \prime }\mathrm{\Phi }Z+\frac{1}{r}{R}^{\prime }\mathrm{\Phi }Z+\frac{1}{r^2}{R\mathrm{\Phi }}^{\prime \prime }Z+{R\mathrm{\Phi }Z}^{\prime \prime }+k_0^2{n}^{2}R\mathrm{\Phi }Z=0$$E26

The solution to the wave equation is deduced from separation of variables, and we obtain:

$$r^2\frac{{\partial }^{2}R}{{\partial r}^2}+r\frac{\partial R}{\partial r}+r^2\left(k_0^2{n}^2-{\beta }^2-\frac{v^2}{r^2}\right)R=0$$E27

The solution is given by Bessel functions: (1),$J_{\nu }\left(\mathit{\kappa r}\right)$ when $k_0^2{n}^2-{\beta }^2-{\nu }^2/r^2$ is positive (${\kappa }^2=k_0^2{n}^2-{\beta }^2$) and (2) $K_{\nu }\left(\mathit{ϓr}\right)$ when $k_0^2{n}^2-{\beta }^2-{\nu }^2/r^2$ is negative (${\gamma }^2={\beta }^2-k_0^2{n}^2$). Bessel function (1) can be approximated by ($\mathit{\kappa r}$ is large) (Figure 9):

$$J_v\left(\mathit{\kappa r}\right)\approx \sqrt{\frac{2}{\mathit{\pi \kappa r}}}\text{cos}\left(\mathit{\kappa r}-\frac{v}{\mathit{\pi r}}-\frac{\pi }{4}\right)$$E28

Figure 9.

Bessel Function of the (a) first kind (behaves as a damped sine wave) and (b) second kind (monotonic decreasing function).

And solution to (2) is

$$K_v\left(\mathit{\gamma r}\right)\approx \frac{e^{-\mathit{\gamma r}}}{\sqrt{2\mathit{\pi \gamma r}}}$$E29

The equation for field distribution in the step-index fiber can be calculated through:

$E_r=\frac{-\mathit{j\beta }}{{\kappa }^2}\left[\mathit{A\kappa }{J}_v^{\prime }\left(\mathit{\kappa r}\right)+\frac{\mathit{\text{jωμv}}}{\mathit{\beta r}}B{J}_v\left(\mathit{\kappa r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}}$, $E_{\varphi }=\frac{-\mathit{j\beta }}{{\kappa }^2}\left[\frac{\mathit{jv}}{r}A{J}_v\left(\mathit{\kappa r}\right)-\frac{\mathit{\omega \mu }}{\beta }\mathit{B\kappa }{J}_v^{\prime }\left(\mathit{\kappa r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}},H_r=\frac{-\mathit{j\beta }}{{\kappa }^2}\left[\mathit{B\kappa }{J}_v^{\prime }\left(\mathit{\kappa r}\right)-\frac{\mathit{j\omega }{ϵ}_{\mathit{\text{core}}}v}{\mathit{\beta r}}A{J}_v\left(\mathit{\kappa r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}}$ and $H_{\varphi }=\frac{-\mathit{j\beta }}{{\kappa }^2}\left[\frac{\mathit{jv}}{r}B{J}_v\left(\mathit{\kappa r}\right)-\frac{\omega {ϵ}_{\mathit{\text{core}}}}{\beta }\mathit{A\kappa }{J}_v^{\prime }\left(\mathit{\kappa r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}}$ for (r<a); a is core’s radius. In the cladding (r>a) $E_r=\frac{\mathit{j\beta }}{{\gamma }^2}\left[\mathit{C\gamma }{K}_v^{\prime }\left(\mathit{\gamma r}\right)+\frac{\mathit{\text{jωμv}}}{\mathit{\beta r}}D{K}_v\left(\mathit{\gamma r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}}$, $E_{\varphi }=\frac{\mathit{j\beta }}{{\gamma }^2}\left[\frac{\mathit{jv}}{r}C{K}_v\left(\mathit{\gamma r}\right)-\frac{\mathit{\omega \mu }}{\beta }\mathit{D\gamma }{K}_v^{\prime }\left(\mathit{\gamma r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}}$ and $H_r=\frac{\mathit{j\beta }}{{\gamma }^2}\left[\mathit{D\gamma }{K}_v^{\prime }\left(\mathit{\gamma r}\right)-\frac{\mathit{j\omega }{ϵ}_{\mathit{\text{clad}}}v}{\mathit{\beta r}}C{K}_v\left(\mathit{\gamma r}\right)\right]e^{\mathit{jv\varphi }}{e}^{-\mathit{j\beta z}}$.

The V-number or the normalized frequency is used to characterize the waveguide and is defined as:

II. Rectangular dielectric Waveguide: Channel/rectangular waveguides are the most commonly used non-planar waveguides for device applications. Channel waveguides include buried waveguides, strip-loaded, ridge, rib, diffused, slot, ARROW, and so on. Figure 10 shows the schematic of few of the channel waveguides. The wave equation analysis of a rectangular waveguide can be done by writing the scalar wave equation:

$$\frac{{\delta }^{2}E}{\delta x^2}+\frac{{\delta }^{2}E}{\delta y^2}+\left[k_0^2{n}^2\left(x,y\right)-{\beta }^2\right]E=0$$E30

$$\text{V}‐\text{number}=a{k}_0\sqrt{n_{\mathit{\text{core}}}^2-n_{\mathit{\text{clad}}}^2}=\frac{2\mathit{\pi a}}{\lambda }\sqrt{n_{\mathit{\text{core}}}^2-n_{\mathit{\text{clad}}}^2}$$E31

Figure 10.

Schematic representation of various channel waveguides.

The general representation of the dielectric waveguide along with the electromagnetic field distribution in the regions is shown below:

$\text{exp}\left(-{\gamma }_{3}x\right)$; $\text{exp}\left(-{\gamma }_{5}y\right)$	$\mathit{Cos}\left({\kappa }_{y}y+{\mathrm{\Phi }}_y\right)$; $\text{exp}\left(-{\gamma }_{3}x\right)$                   3	$\text{exp}\left(-{\gamma }_{3}x\right)$; $\text{exp}\left(-{\gamma }_4\left(y-b\right)\right)$
$\mathit{Cos}\left({\kappa }_{x}x+{\mathrm{\Phi }}_x\right)$; $\text{exp}\left({\gamma }_{5}x\right)$                      5	$\mathit{Cos}\left({\kappa }_{x}x+{\mathrm{\Phi }}_x\right)$; $\mathit{Cos}\left({\kappa }_{y}y+{\mathrm{\Phi }}_y\right)$            1	$\mathit{Cos}\left({\kappa }_{x}x+{\mathrm{\Phi }}_x\right)$; $\text{exp}\left(-{\gamma }_4\left(y-b\right)\right)$           4
$\text{exp}\left(-{\gamma }_2\left(x-a\right)\right)$; $\text{exp}\left({\gamma }_{5}y\right)$	$\mathit{Cos}\left({\kappa }_{y}y+{\mathrm{\Phi }}_y\right)$; $\text{exp}\left(-{\gamma }_2\left(x-a\right)\right)$         2	$\text{exp}\left(-{\gamma }_4\left(y-b\right)\right)$; $\text{exp}\left(-{\gamma }_2\left(x-a\right)\right)$

where ${\varphi }_x$ and ${\varphi }_y$ are phase constants. The characteristic equations are given by $\mathit{tan}{\kappa }_{y}b=\frac{{\kappa }_y\left({\gamma }_4+{\gamma }_5\right)}{{\kappa }_y^2-{\gamma }_4{\gamma }_5}$ and $\mathit{tan}{\kappa }_{x}a=n_1^2\frac{{\kappa }_x\left(n_2^2{\gamma }_3+n_3^2{\gamma }_2\right)}{n_2^2{n}_3^2{\kappa }_x^2-n_1^2{\gamma }_2{\gamma }_3}$ (${\gamma }_i$) are exponential decay constants. The critical cut-off condition is given by:

$$V=k_0\frac{a}{2}\sqrt{n_1^2-n_2^2}$$E32

The following section describes various types of channel waveguides.

1. Wire waveguide: The schematic of silicon photonic wire waveguide is shown in Figure 11(a). The waveguide consists of a silicon core and silica-based cladding. Since the single-mode condition is very important in constructing functional devices, the core dimension should be determined so that a single-mode condition is fulfilled. The primary requisite is single-mode guiding of the TE₀₀ and TM₀₀ mode. When the effective refractive index is larger than the cladding and smaller than the core, mode is guided in the waveguide, and guiding will be stronger for higher values of effective index $n_{\mathit{eff}}.$ Thus, modes with effective indices above $n_{S{\mathit{iO}}_2}$ will not be radiated into the buffer layer and thus will be guided. Figure 11(b) depicts the quasi-TE mode of a 220-nm-high and 450-nm-wide silicon waveguide at wavelength of 1.55 μm [8].

Figure 11.

(a) Silicon-on-insulator wire waveguide, (b) quasi-TE mode of a 220-nm-high and 450-nm-wide silicon waveguide at wavelength of 1.55 μm, and (c) effective refractive index ($n_{\mathit{eff}})$ at 1550 nm for a 220-nm-high silicon photonic wire waveguide. The left of the hashed line is the single-mode region.

Each mode propagates through the waveguide with a phase velocity of $c/n_{\mathit{eff}}$, where c denotes the speed of light in vacuum and $n_{\mathit{eff}}$ is the effective refractive index felt by that mode. It signifies how strongly the optical power is confined to the waveguide core. Most waveguides support modes of two independent polarizations, with either the major magnetic (quasi-TM) or electric (quasi-TE) field component along the transverse (horizontal) direction.

Figure 11(c) shows $n_{\mathit{eff}}$ as a function of the width of the photonic wire. The $n_{\mathit{eff}}$ depends on the waveguide cross-section, waveguide materials, and the cladding material. Higher-order modes travel with a different propagation constant compared to the lowest-order mode and are less confined in the waveguides. As a consequence of the dissimilar propagation constants, there is modal dispersion which reduces the distance-bandwidth product of the waveguide. Due to the low confinement, first, a large field decay outside the waveguide reduces the maximum density of the devices and, second, in the waveguide bends the higher-order modes become leaky resulting in propagation losses. It is desirable that the difference between $n_{\mathit{eff}}$ of the fundamental quasi-TE and quasi-TM modes be large so that the coupling between the modes is limited due to difference in mode profiles and also the phase-mismatch. For widths below ∼550 nm, silicon photonic wire will be single mode for each polarization.

2. Rib waveguide: Figure 12(a) and (b) shows the schematic and the fundamental quasi-TE mode of a silicon photonic rib waveguide (H = 220 nm, r = 70 nm). Although a rib waveguide can never truly be single mode, by optimizing the design, the power carried by the higher-order modes will eventually leak out of the waveguide over a very short distance, thus leaving only the fundamental mode. Figure 12(c) shows the dispersion $\left(n_{\mathit{eff}}\right)$ as a function of the width of the photonic rib waveguide. For widths below ∼800 nm, silicon photonic rib waveguide will be single mode for each polarization.

Figure 12.

(a) Silicon-on-insulator rib waveguide, (b) quasi-TE mode of a 220-nm-high and 700-nm-wide silicon rib waveguide at wavelength of 1.55 μm, and (c) effective refractive index ($n_{\mathit{eff}})$ at 1550 nm for 220-nm-high silicon rib waveguide for ridge height (r) = 70 nm.

Wire waveguides are advantageous as they provide a small bending radius and realization of ultra-dense photonic circuits. However, they have higher propagation losses. On the one hand, wire waveguide allows low-loss sharp bends in the order of a few micrometres, while, on the other hand, the device structures produced are susceptible to geometric fluctuations such as feature drift size (resulting in degradation of device performance) and waveguide sidewall roughness (resulting in propagation losses) [9, 10]. Rib waveguides typically require bend radii >50 μm in SOI to ensure low bend losses, which eventually result in a larger device/circuit footprint. Figure 13 shows the TE mode loss in silicon wire and rib waveguide for a bend of 90^°.

Figure 13.

Mode loss for silicon wire (cross-section: 450 × 220 nm²) and rib (cross-section: 600 × 220 nm²) waveguides for a 90^° bend with increasing bending radii.

3. Slot waveguide

Slot waveguides are used to confine light in a low-index material between two high-index strip waveguides by varying the gap and dimensions (width and height) of the strip waveguides (Figure 14(a)). The normal component of the electric field (quasi TE) undergoes very high discontinuity at the boundary between a high- and a low-index material, which results into higher amplitude in the low-index slot region. The amplitude is proportional to the square of the ratio between the refractive indices of the high-index material (Si, Ge, Si₃N₄) and the low-index slot material (air). On the other hand, the effect of the presence of the slot is minimal on quasi-TM mode, which is continuous at the boundary. When the width of the slot waveguides is comparable to the decay length of the field, electric field remains across the slot and the section has high-field confinement [11, 12, 13, 14], which results into propagation of light in the slot section; unlike in a conventional strip waveguide, where the propagating light is confined mainly in the high-index medium.

Figure 14.

(a) Silicon-on-insulator slot waveguide, (b) Quasi-TE mode of a 220-nm-high (Gap = 100 nm) slot waveguide, (b) variation of effective refractive index with waveguide width for slot gap of 100, 150, and 200 nm, respectively, at a wavelength of 1.55 μm.

Figure 14(c) shows the variation in effective index with the waveguide width for different slot gaps. The advantage of a slot waveguide is the high-field confinement in the slot section, which normally cannot be achieved using a simple strip- or a ridge-based waveguide, making it a potential candidate for applications that require light-matter interaction such as sensing [12] and nonlinear photonics [13]. The launching of light into a slot waveguide is normally done by phase matching the propagation constant of the strip waveguide and the slot waveguide. However, efficient coupling still remains a challenge because of scattering loss and mode mismatch of the slot and strip waveguides, with a reported propagation loss between 2 and 10 dB/cm [14].

4. Strip-loaded waveguide

A strip-loaded waveguide is formed by loading a planar waveguide, which already provides optical confinement in the x direction, with a dielectric strip of index $n_3<n_1$ or a metal strip to facilitate optical confinement in the y direction, as shown in Figure 15(a). Strip-loaded waveguides do not require half-etching in waveguide fabrication and is therefore easier to fabricate. Figure 15(a) shows the schematic of a hydrogenated amorphous silicon strip-loaded waveguide where a thermal oxide is inserted between the layers for passivation [15]. Figure 15(b) shows the optical field for the waveguide for a 75-nm-thick and 800-nm-wide strip-loaded waveguide and Figure 15(c) depicts the variation in effective index with the strip waveguide width.

Figure 15.

(a) Hydrogenated amorphous strip-loaded waveguide, (b) Quasi-TE mode of a 220-nm-high, 800-nm-wide, 75-nm-thick strip Waveguide, (c) Variation of effective refractive index with strip width at a wavelength of 1.55 μm.

5. Suspended waveguide

Suspended waveguides have enabled new types of integrated optical devices for applications in optomechanics, nonlinear optics, and electro-optics. Fabrication involves removing a sacrificial layer above or below a waveguide core layer to design these waveguides [16]. Increasing absorption loss of ${\mathit{SiO}}_2$ at longer wavelengths makes it challenging to utilize SOI for low-loss components in the mid-infrared (MIR) [17]. Removing the ${\mathit{SiO}}_2$ layer opens the possibility of extending the low-loss SOI wavelength range up to ∼8 μm. For MEMS, it is imperative to have waveguides that can be mechanically actuated. This requires waveguides that are released from the substrate, for example, through surface micromachining [17]. Figure 16 shows the schematic of a suspended waveguide [18].

Figure 16.

Schematic of a suspended waveguide.

6. Triplex^TM technology

TriPleX waveguides are a family of waveguide geometries that is based on an alternating layer stack consisting of two materials: ${\mathit{Si}}_3{N}_4$ and ${\mathit{SiO}}_2$. The waveguide geometries are categorized as box shell, single stripe (propagation loss <0.03 dB/cm), symmetric double stripe (propagation loss <0.1 dB/cm), and asymmetric double stripe (propagation loss <0.1 dB/cm) as shown in Figure 17(a) [19]. Different confinement regimes can be optimized for specific applications for these waveguides and tunable birefringence- and polarization dependent loss (PDL) can be achieved. Propagation losses (<0.1 dB/cm), very low PDL (< 0.1 dB/cm), and easy interconnection with optical fibers (<0.15 dB/facet) have been demonstrated in single-mode box-shaped waveguides [20]. Moreover, fabrication of the waveguide is a low-cost and simple process.

Figure 17.

(a) Schematic of different type of TriPleX waveguides, (b) Variation in waveguide size of the box-shaped waveguide, and (c) its diffraction angle versus the index contrast.

LioniX TriPleX technology is a versatile photonics platform suited for applications such as communications, biomedicine, sensing, and so on, over a broadband range of 0.4 to 2.35 μm [21]. Figure 17(b) and (c) depicts the variation in waveguide size and diffraction angle with index contrast of the box-shaped geometry for a wavelength of 1.55 μm.

7. Photonic crystal waveguide

Photonic crystal waveguides guiding mechanism is different from that of a traditional waveguide, which is based on internal reflection. A photonic crystal is a periodic dielectric structure with a photonic band gap, that is, a frequency range over which there is no propagation of light. The introduction of line defects into a photonic crystal structure creates an optical channel for propagation of light. If the line defect is properly designed, the resulting guiding mode falls within a photonic band gap, is highly confined, and can be used for guiding light. The guiding mode can also be designed to be broadband and thus gives rise to a compact, broadband photonic crystal waveguide [22]. Application of these waveguides includes nanofluidic tuning, RI measurements, optical characterization of molecule orientation, and biosensing.

8. Diffused waveguide

A diffused waveguide is formed by creating a high-index region in a substrate through diffusion of dopants, such as a LiNbO₃ waveguide with a core formed by Titanium (Ti) diffusion. Due to the diffusion process, the core boundaries in the substrate are not sharply defined. A diffused waveguide has a thickness defined by the diffusion depth of the dopant and a width defined by the distribution of the dopant. Alternatively, the material can be exchanged with the substrate. Ion-exchanged glass waveguide is fabricated by diffusing mobile ions originally in glass with other ions of different size and polarizability [23].The additional impurities cause a change in refractive index that is approximately proportional to their concentration. A material can also be implanted using an ion implanter within the waveguide. However, this process damages the lattice and is therefore followed by annealing.

9. ARROW waveguide

In anti-resonant reflecting optical (ARROW) waveguides, light confinement is realized by choosing the cladding layer thicknesses accordingly to create an anti-resonant Fabry-Perot reflector for the transverse component of the wave vector at the desired wavelength. Even though the ARROW mode is leaky, low-loss propagation over large distances can be achieved. Yin et al. have designed an ARROW waveguide exhibiting single-mode confinement and low-loss light propagation in a hollow air core on a semiconductor chip [24]. ARROW waveguides with non-solid low-index cores have applications in gas and liquid sensing, quantum computing, quantum communications, and Raman scattering spectroscopy. Chalcogenide rib ARROW structures have also been shown with propagation loss ∼6 dB/cm to design opto-chemical sensors in the near- and mid-IR region [25].

10. Augmented waveguide

Light confinement in a low-index media has been shown in ARROW, slot, and plasmonic waveguides. However, ARROW waveguide has low confinement and is thus leaky. Strong light confinement in the low-index medium can be achieved by using silicon slot and plasmonic waveguide. Fabrication of the slot waveguide is cumbersome and hybrid plasmonic waveguide suffers from additional propagation losses due to the presence of metal. Augmented waveguide confines light efficiently in the low-index region by reducing the reflection at the high index-low index interface in a high-index contrast waveguide, which results in enhancement of light confinement in the low-index region [26]. Figure 18 shows the schematic of an augmented low-index waveguide.

Figure 18.

Schematic of an augmented waveguide.

Waveguides can be classified on the basis of different material platforms. Wavelength range, ease of fabrication, compactness, and CMOS compatibility are few of the determining factors when selecting a material for a specific application. Table 1 compares various waveguide platforms along with their propagation losses [27]. Figure 19 shows variation of index contrast with footprint for few material platforms.

Material platforms	Waveguides	Range	Configuration	Propagation Loss
Semiconductor materials	Silicon	Mid-IR	Silicon nanophotonic waveguide	∼4 dB/cm (2030 nm); ∼10 dB/cm (2500 nm) [28]
		Mid/Near-IR	Suspended silicon-membrane ridge waveguide (TM mode)	∼$2.8\pm 0.5$ dB/cm (3.39 μm); ∼$5.6\pm 0.3$ dB/cm (1.53 μm) [29]
		Mid/Near-IR	Silicon on porous silicon (SiPSi)	∼$2.1\pm 0.2$ dB/cm (1.55 μm) ∼$3.9\pm 0.2$ dB/cm (3.39 μm) [30]
	Germanium	Mid-IR	Germanium on silicon strip waveguide	$\sim 2.5-3\mathrm{dB}$ dB/cm [31] (∼5.15-5.4 μm)
		Far-IR	Germanium on silicon strip waveguide	$\sim 2.5\mathrm{dB}$ dB/cm [32] (∼5.8 μm)
		Mid-IR	Germanium on silicon rib waveguide	$\sim 2.4\pm 0.2$ dB/cm [33] (∼3.8 μm)
		Mid-IR	Germanium-rich silicon germanium platform-based rib waveguide	$1.5\pm 0.5$ dB/cm [34] (∼4.6 μm)
		Mid IR/Far-IR	Silicon germanium/silicon-based graded index waveguides	∼$1$ dB/cm (4.5 μm) ∼$2$ dB/cm (7.4 μm) [35]
		Mid-IR	Germanium-on-silicon-on-insulator waveguides	$3.5$ dB/cm (3.682 μm) [36]
		Mid-IR	Silicon germanium ridge waveguide on a silicon substrate	$0.5$ dB/cm (4.75 μm) [37]
		Far-IR	Germanium on GaAs ridge waveguide	4 $.2$ dB/cm (10 μm) [38]
	Gallium arsenide (GaAS)	Near-IR	GaAS/Al0.3 Ga0.7As ridge waveguide for manipulation of single-photon and two-photon states	$1.6$ dB/cm (1.55 μm) [39]
		Near-IR	Suspended GaAs waveguide	$0.4$ dB/mm (TE) (1.55 μm) and 6 dB/mm (TM) (1.03 μm) [40]
		Near-IR	GaAs-based single-line defect photonic crystal slab waveguide	$0.76$ dB/mm (1050–1580 nm) [41]
	Indium phosphide (InP)	Near-IR	InP waveguides based on localized Zn-diffusion process (MOVPE) to mitigate passive loss by p-dopants	0.4 dB/cm (1.55 μm) [42]
		Near-IR	Suspended InP dual-waveguide structures for MEMS-actuated optical buffering	2.2 dB/cm (1.5–1.6 μm) [43]
	Gallium antimony	Mid-IR	GaSb waveguides based on quasi-phase matching (QPM)	∼$0.7/1.1$ dB/cm [44] ∼(2/4 μm)
	Quantum dotes	Near-IR	Polymer waveguides containing infrared-emitting nanocrystal quantum dots (PbSe and InAs)	∼$5$ dB/cm (inclusive of fiber coupling loss) [45] (∼1550 nm)
	Doped semiconductor	Vis/Near-IR	Rare-earth-doped GaN (gallium nitride) channel waveguide	∼$5.4/4.1$ dB/cm [46] (∼633/1550 nm)
		Near-IR	Erbium-Doped Phosphate Glass Strip-loaded Waveguide on Silicon	∼$4.1$ dB/cm [47] (∼1535 nm)
	Semiconductor nanomaterials	Mid-IR	Silicon-on-Sapphire Suspended Nanowire	∼$1\pm 0.3$ dB/cm [48] (∼4 μm)
		Vis	Tin-oxide nanoribbons-based subwavelength waveguide	∼$10$ dB/cm [49] (∼400-550 nm)
		Near-IR	Amorphous silicon nanowire	∼$4.5$ dB/cm [50] (∼1550 nm)
		Near-IR	InP/benzocyclobutene optical nanowires on a GaAs substrate	∼$0.8$ dB/mm [51] (∼1550 nm)
Silicon-on-insulator waveguides	Silicon-on-Silica	Near-IR	Silicon-on-silica strip waveguide	∼$0.6$ dB/cm [52] (∼1550 and 2000 nm)
		Near-IR	Silicon-on-silica rib (70-nm etch depth) waveguide	∼$0.1-0.2$ dB/cm [52] (∼1550 and 2000 nm)
		Mid-IR	Silicon-on-silica rib (70-nm etch depth) waveguide	∼$1.5$ dB/cm [53] (∼3800 nm)
		Near-IR	Silicon-on-silica slot waveguide	∼$2.28\pm 0.03$ dB/cm [54] (∼1064 nm)
		Near-IR	Silicon-on-Silica slot waveguide	∼$3.7$ dB/cm [55] (∼1550 nm)
		Mid-IR	Suspended silicon waveguide	∼3.1 dB/cm (7.67 μm) [56]
		Near-IR	Silicon-on-silica strip waveguide coated with amorphous $\left({\mathrm{TiO}}_2\right)$	∼$2\pm 1$ dB/cm [57] (∼1550 nm)
	Silicon-on-Sapphire	Mid-IR	Silicon-on-sapphire ridge waveguide	∼$4.0\pm 0.7$ dB/cm [58] (∼5.4-5.6 μm)
		Mid-IR	Silicon-on-sapphire ridge waveguide	∼$4.3\pm 0.6$ dB/cm [59] (∼4.5 μm)
		Mid-IR	Silicon-on-sapphire slot waveguide	∼$11$ dB/cm [60] (∼3.4 μm)
		Near/Mid-IR	Silicon-on-sapphire nanowire waveguide	∼$0.8$ dB/cm [61] (∼1550 nm).; ∼$1.1-1.4$ dB/cm [61] (∼2080 nm); ∼$<2$ dB/cm [61] (∼5.18 μm)
	Silicon-on-nitride	Mid-IR and Near-IR	Silicon-on-nitride ridge waveguide	∼$5.2\pm 0.6$ dB/cm [62] (∼3.39 μm)
	Thallium-doped SOI Rib/Indium-doped SOI Rib	Near-IR	Thallium-doped silicon waveguide	∼$3$ dB/cm [63] (∼1.55 μm)
		Near-IR	Indium-doped silicon waveguide, decrease in absorption coefficient ∼$16$ dB/cm [64] (wavelength ∼1.55 μm).
Glass waveguides	Silica glass	Vis	Laser-written waveguide in fused silica for vertical polarization (VP)/horizontal polarization (HP) beam	∼$0.06/0.1$ dB/cm [65] (∼777 nm)
		Mid-IR	3D laser-written silica glass step-index high-contrast (HIC) waveguide	∼$1.3$ dB/cm^3 [66] (∼3.39 μm)
		Near-IR	Graded-index (GRIN) Cladding in HIC glass waveguides	∼$1.5$ dB/cm [67] (∼1.55 μm)
		Near-IR	High-index, doped silica glass material (Hydex) waveguides	∼$0.06$ dB/cm [68] (∼1.55 μm)
	Silicon oxynitride (SiON)	Near-IR	SiON deposited by inductively coupled PECVD-based waveguides	∼$0.5\pm 0.05$ dB/cm,$1.6\pm 0.2$ dB/cm and $0.5\pm 0.06$ dB/cm [69] (∼1330, 1550 and 1600 nm)
	Ion—exchanged glass	Vis	Ti^+/Na^+ ion-exchanged single-mode waveguides on silicate glass	∼$9\pm 1$ dB/cm [70] (∼0.405 μm)
		Near-IR	Alkaline aluminum phosphate glasses for thermal ion-exchanged waveguide	∼$0.53$ dB/cm [71] (∼1.534 μm)
	Sol-gel glass	Near-IR	Hybrid organic-inorganic glass sol-gel ridge waveguides	∼$0.1$ dB/cm [72] ∼(1.55 μm)
		Vis	Sol-gel derived silicon titania slab waveguides films	∼$0.2$ dB/cm [73] ( 677 nm)
		Vis	Sol–gel-derived glass-ceramic photorefractive films-based waveguide	∼$0.5\pm 0.2$ dB/cm [74] (∼635 nm)
	Tungsten tellurite glass	Near-IR	Optical planar channel waveguide-based on tungsten-tellurite glass fabricated by RF Sputtering	∼$0.44$ dB/cm [75] (∼1.53 μm)
	Laser-written	Near-IR	Laser-written waveguide in planar light-wave circuit (PLC) glass doped with Boron and Phosphorous	∼$0.35$ dB/cm [76] (∼1.55 μm)
		Vis	Laser-written waveguide in fused silica for vertical polarization (VP)/horizontal polarization (HP) beam	∼$0.06/0.1$ dB/cm [65](∼777 nm)
		Near-IR	Laser-written ferroelectric crystal in glass waveguide	∼$2.64$ dB/cm [77] (∼1530 nm)
		Vis	Femtosecond laser-written double-line waveguides in germanate and tellurite glasses	∼$2.0$ dB/cm [78] (∼632 nm)
		Near-IR	Ultrafast laser-written waveguides in flexible As2S3 chalcogenide glass tape	<$0.15$ dB/cm [79] (∼1550 nm)
Electro-optic waveguides	Lithium niobate	Near-IR	Lithium niobate on insulator rib waveguide	∼$0.4$ dB/cm [80] (∼1.55 μm)
		Near-IR	Lithium niobate ridge waveguide	∼$0.3$ dB/cm(TE) and $0.9$ dB/cm(TM) [81] (∼1.55 μm)
		Near-IR	Periodically poled lithium niobate waveguide	∼$<1$ dB/cm [82] (∼1.55 μm)
		Near-IR	Heterogeneous lithium niobate on silicon nitride waveguide	∼$<0.2\pm 0.4$ dB/cm [83] (∼1.54 μm)
		Near-IR	Lithium Niobate on Insulator Ridge Waveguide	∼$1$ dB/cm [84] (∼1.55 μm)
		Near-IR	Thin film strip-loaded (SiN) lithium niobate waveguide	∼$5.8$ dB/cm (TE)$14$ dB/cm (TM) [85] (∼1.55 μm)
		Near-IR	Thin film strip-loaded (a-Si) lithium niobate waveguide	∼$42$ dB/cm (TE)$20$ dB/cm (TM) [86] (1.55 μm)
		Near-IR	Thin film lithium niobate ridge waveguide	∼$0.268$ dB/cm (TE)$1.3$ dB/cm (TM) [87]; (∼1.55 μm)
	Lithium tantalate	Near-IR	Ga^3+-diffused lithium tantalate waveguide	∼$0.2$ dB/cm (TE)$0.4$ dB/cm (TM) [88] (∼1.55 μm)
	Barium titanate (BTO)	Vis	BTO thin films on MgO-based ridge waveguide	∼$2$ dB/cm (TE) [89] (∼633 nm)
		Mid-IR	BTO thin films on Lanthanum Aluminate (LAO) with a-Si ridge waveguide	∼$4.2$ dB/cm [90] (∼3.0 μm)
	Electro-optic polymer	Near-IR	Hybrid Electro-Optic Polymer and TiO2 Slot Waveguide	∼$5$ dB/cm [91] (∼1550 nm)
	Liquid crystal	Near-IR	PDMS (poly (dimethyl siloxane))-liquid crystal-based optical waveguide	∼$8$ dB/cm [92] (∼1550 nm)
		Vis	Liquid-crystal core channel waveguide encapsulated in semicircular grooves with glass substrate	∼$1.3$ dB/cm [93] (∼632.8 nm)
		Near-IR	Liquid crystal clad shallow-etched SOI waveguide	∼$4.5$ dB/cm [94] (1570 nm)
Polymer-based	Conventional optical polymers	Vis	PMMA (poly(methyl methacrylate) )-based optical waveguide	∼$0.2$ dB/cm [95] (∼850 nm)
		Vis/Near-IR	Polyurethane (PU)-based optical waveguide	∼$0.8$ dB/cm [95] (∼633 and 1064 nm)
		Vis/Near-IR	Epoxy resin-based optical waveguide	∼$0.3/0.8$ dB/cm [95] (∼633/1064 nm)
		Vis	Polymer PMMA-based waveguide using femtosecond laser	∼$0.3$ dB/cm [96] (∼638 and 850 mm)
	Novel optical polymers	Vis/Near-IR	Polymeric waveguides (WIR30 photopolymer) with embedded micro-mirror	∼$0.18$ dB/cm [97] (∼850 nm )
		Vis/Near-IR	Acrylate-based waveguide pattern using photo exposure/laser ablation	∼$0.02/0.$ 3 and 0.8 dB/cm [95] (∼840/1300 and 1550 nm)
		Vis/Near-IR	Telephotonics-OASIC-based optical waveguide	∼$<0.01,0.03,0.1$ dB/cm [95]; (∼840/1300/1550 nm)
		Near-IR	Dow chemical perfluorocyclobutane(XU 35121)-based waveguide	∼$0.25$ dB/cm [95] (∼1300/1550 nm)
		Near-IR	Circular-core UV-curable epoxies-based optical waveguide	∼$0.79$ dB/cm [98] (∼1550 nm)
		Vis/Near-IR	Multi-mode Siloxane-based polymer waveguide, single-mode siloxane-based polymer waveguide	∼$0.05$ dB/cm [99] (∼850 nm)/ ∼$0.5/1$ dB/cm [99] (∼1.31/1.55μm)
	Surface plasmon polariton waveguide	Near-IR	Polymer-Silicones-based long range surface plasmon polariton waveguide (LRSPPW)	∼$0.5-1$ dB/mm [100] (∼1.55 μm)
		Near-IR	Exguide ZPU/LFR-based long range surface plasmon polariton waveguide (LRSPPW)	∼$1.72$ dB/cm [101] (∼1.55 μm).
Hollow waveguides	Metal/Dielectric Coated	THz	Gold-coated waveguide using liquid phase chemical deposition process	∼$1.98/1.89$ dB/cm [102] (∼215/513 μm)
		THz	Silver-coated waveguide using liquid phase chemical deposition Process	∼$1.77/1.62$ dB/cm [102] (∼215/513 μm)
		Mid-IR	Cadmium sulfide CdS) and PbS (Lead Sulfide) on Ag (Silver)-coated hollow glass waveguide	∼$0.037$ dB/cm [103] (∼2.94 μm)
		THz	High-refractive index composite photonics band-gap Bragg fiber	∼$0.3$ dB/cm [104] (∼300 μm)
		THz	Silver/polystyrene-coated hollow glass waveguides	∼$1.9$ dB/m [105] (∼300 μm)
		Vis/Near-IR	Silver/cyclic olefin hollow glass waveguide	∼$0.549/0.095\text{and}0.298$ dB/m [106]; (∼808/1064 and 2940 nm)
	Hollow glass	Near-IR	Hollow-core optical waveguide (Si Substrate)	∼$1.7$ dB/cm [107] (∼1.52-1.62 μm)
		Far-IR	Tapered hollow-air core waveguide	∼$1.27$ dB/cm [108] (∼6.2 μm)
		Vis	Air-core anti-resonant reflecting Optical Waveguide (ARROW)	∼$4$ dB/cm [109] (∼820-880 nm)
	Chalcogenide	Near-IR	Chalcogenide waveguides (${\mathit{Ge}}_{23}{\mathit{Sb}}_7{S}_{70}$) fabricated through CMOS-compatible lift-off process: Strip/rib	2-6 /<0.5 dB/cm [110] (∼1550 nm)
		Mid-IR	Chalcogenide waveguides (${\mathit{Ge}}_{11.5}{\mathit{As}}_{24}{\mathit{Se}}_{64.5}$)	<1 dB/cm (0.3dB at 2000 cm^−1 ); (1500-4000 cm^−1) [111]
		Near-IR	Chalcogenide waveguides ${\mathit{Ge}}_{23}{\mathit{Sb}}_7{S}_{70}$ (chlorine-based plasma etching)	<0.42 dB/cm^−1 (1550 nm) [112]
	Liquid core	Vis	Liquid core/(ethylene glycol) air-cladding waveguide	∼$0.14$ dB/cm ( 464-596 nm) [113]
		Vis	Nano porous solid liquid core Waveguide	∼$0.6$ dB/cm ( 632.8 nm) [114]
Metamaterial optical waveguides		Mid-IR	Suspended silicon waveguide with lateral cladding (subwavelength grating metamaterial)	∼0.82 dB/cm (3.8μm) [115]
Titanium dioxide $\left({\mathrm{TiO}}_2\right)$		Vis/Near-IR	Atomic layer deposition (ALD) ${\mathrm{TiO}}_2$ slab waveguide	∼2.0-3.5 dB/cm (633 nm) [116]; <1 dB/cm (1530 nm) [65]
		Near-IR	Amorphous ${\mathrm{TiO}}_2$ strip waveguide	∼2.4-0.2 dB/cm (1.55μm) [117]
Silicon carbide		Near-IR	PECVD silicon-carbide-silicon oxide horizontal slot waveguide	∼$23.9\pm 1.2$ dB/cm (1.3 μm)-TM mode [118]
Silicon nitride on silica^30		Vis	Silicon-nitride strip waveguide	∼$2.25$ dB/cm. [119] (∼532 nm)
		Vis	Silicon-nitride strip waveguide	∼$0.51$ dB/cm. [120] (∼600 nm)
		Vis	Silicon-nitride strip waveguide	∼$1.30$ dB/cm. [119] (∼780 nm)
		Near-IR	LPCVD Silicon-nitride strip waveguide	∼$0.04$ dB/cm. [121] (∼1550 nm)
		Near-IR	TripleX ^TM LPCVD silicon-nitride planar waveguide	∼$0.02$ dB/cm. [122] (∼1550 nm)
		Near-IR	900-nm-thick LPCVD Silicon-nitride strip waveguide	∼$0.37$ dB/cm. [123] (∼1550 nm)
		Mid—IR	LPVCD silicon-nitride strip waveguide	∼$0.60$ dB/cm. [124] (∼2600 nm)
		Mid-IR	Silicon-rich LPVCD silicon-nitride strip waveguide	∼$0.16$ dB/cm. [125] (∼2650 nm)/$2.10$ dB/cm. [126] (∼3700 nm)
Tantalum pentoxide (${\mathit{Ta}}_2{O}_5$)-core/silica-clad/silicon substrate		Near IR	Planar waveguide	∼$0.03$ dB/cm. [127] (∼1550 nm)
Suspended silicon-on-insulator waveguide		Mid-IR	Waveguide with Subwavelength grating	∼$3.4$ dB/cm. [17] (∼3.8 μm).
Photonic crystal fibers based waveguides		THz	Semiconductor silicon photonic crystal slab waveguides	∼$0.1$/0.04 dB/cm. [128] (∼905.7/908 μm).
		THz	3-D printed THz waveguide based on Kagome photonic crystal structure	∼$0.1$/0.04 dB/cm. [129] (∼905.7/908 μm).
		THZ	Kagome-lattice hollow-core silicon photonic crystal slab-based waveguide	∼$0.875$ dB/cm [129] (∼400 μm).

Table 1.

Material platforms.

Figure 19.

Comparison of different waveguide platforms as a function of index contrast and compactness.

11. Conclusion

Classification of waveguides on the basis of geometry (planar/non-planar), mode propagation (Single/Multi-Mode), refractive index distribution (Step/Gradient Index), and material platform is described briefly. An overview of different kinds of channel waveguides, namely wire, rib, slot, strip-loaded, diffused, TriPleX, suspended, photonic crystal, ARROW, and augmented waveguide is given. A comparative analysis of material platforms used along with their propagation losses and wavelength range is also shown.