Bent waveguide model

Consider a bent slab waveguide with the $y$-axis as the axis of symmetry as shown in Figure 2.1. We assume that the material properties and the fields do not vary in the $y$-direction. Being specified by the radially dependent refractive index $n(r)$ (here $n$ is piecewise constant), the waveguide can be seen as a structure that is homogeneous along the angular coordinate $\theta$. Hence one chooses an ansatz for the bend modes with pure exponential dependence on the azimuthal angle, where the angular mode number is commonly written as a product $\gamma R$ with a reasonably defined bend radius $R$, such that $\gamma$ can be interpreted as a propagation constant.

Figure 2.1: A bent slab waveguide. The core of thickness $d$ and refractive index $n_{\mbox{\scriptsize f}}$ is embedded between an interior medium (``substrate'') with refractive index $n_{\mbox{\scriptsize s}}$ and an exterior medium (``cladding'') with refractive index $n_{\mbox{\scriptsize c}}$. The distance between the origin and the outer rim of the bend defines the bend radius $R$.

In the cylindrical coordinate system $(r,y,\theta)$, the functional form (in the usual complex notation) of the propagating electric field $\boldsymbol {E}$ and the magnetic field $\boldsymbol {H}$ reads

$$\begin{aligned}\boldsymbol {E}(r, \theta, t) & = (\tilde{E}_{r}... ...displaystyle \mbox{i}(\omega t - \gamma R \theta)}, \end{aligned}$$

where the $\sim$ symbol indicates the mode profile, $\gamma$ is the propagation constant of the bend mode, and $\omega$ is the angular frequency corresponding to vacuum wavelength $\lambda$. Since an electromagnetic field propagating through a bent waveguide loses energy due to radiation [88], $\gamma$ is complex valued, denoted as $\gamma = \beta -$   i$\alpha$, where $\beta$ and $\alpha$ are the real valued phase propagation and attenuation constants.

Note that the angular behaviour of the field (2.1) is determined by the product $\gamma R$, where the definition of $R$ is entirely arbitrary. Given a bend mode, the values assigned to the propagation constant $\gamma$ change, if the same physical solution is described by using different definitions of the bend radius $R$. We will add a few more comments on this issue in Section 2.4. The definition of the bend radius $R$ as the radial position of the outer interface of the core layer is still applicable in case the guiding is effected by a single dielectric interface only, i.e. for the description of whispering gallery modes (see Section 2.4.5). Hence, for this paper we stick to the definition of $R$ as introduced in Figure 2.1.

If the ansatz (2.1) is inserted into the Maxwell equations, one obtains the two separate sets of equations

$$\left. \begin{aligned}\frac{\gamma R}{r}\tilde{E}_{y} & = -\mu_... ...}\epsilon_0 \epsilon \omega \tilde{E}_{y} \end{aligned} \right \}$$

and

$$\left. \begin{aligned}\frac{\gamma R}{r}\tilde{H}_{y} &= \epsil... ... &= \mbox{i}\mu_{0} \omega \tilde{H}_{y}, \end{aligned} \right \}$$

with vacuum permittivity $\epsilon_0$, vacuum permeability $\mu_0$, and the relative permittivity $\epsilon = n^2$.

For transverse electric (TE) waves the only nonzero components are $\tilde{E}_{y}$, $\tilde{H}_{r}$ and $\tilde{H}_{\theta}$, which are expressed in terms of $\tilde{E}_{y}$, while for transverse magnetic (TM) waves the only nonzero components are $\tilde{H}_{y}$, $\tilde{E}_{r}$ and $\tilde{E}_{\theta}$, which are given by $\tilde{H}_{y}$. Within radial intervals with constant refractive index $n$, the basic electric and magnetic components are governed by a Bessel equation with complex order $\gamma R$,

$$\frac{\partial ^{2} \phi }{\partial r^{2}} + \frac{1}{r} \, \frac... ...tial \phi}{\partial r} + (n^{2} k^{2} - \frac{\gamma^{2} R^{2}}{r^2} ) \phi = 0$$ (2.4)

for $\phi = \tilde{E}_{y}$ or $\phi = \tilde{H}_{y}$, where $k=2 \pi / \lambda$ is the (given, real) vacuum wavenumber. For TE modes, the interface conditions require continuity of $\tilde{E}_{y}$ and of $\partial_r \tilde{E}_{y}$ across the dielectric interfaces. For TM modes, continuity of $\tilde{H}_{y}$ and of $\epsilon^{-1} \partial_r \tilde{H}_{y}$ across the interfaces is required.

Eq. (2.4), together with the interface conditions and suitable boundary conditions for $r\rightarrow 0$ and $r \rightarrow \infty$, represents an eigenvalue problem with the bend mode profiles $\phi$ as eigenfunctions, and the propagation constants $\gamma$ or angular mode numbers $\nu = \gamma R$ as eigenvalues. The equation is solved piecewise in the regions with constant refractive index. While the procedure is in principle applicable for arbitrary multilayer bent waveguides, for the sake of brevity we discuss here the three layer configuration as introduced in Figure 2.1.

The general solution of Eq. (2.4) is a linear combination of the Bessel functions of the first kind J and of the second kind Y. This representation is applicable to the core region. Since Y tends to $-\infty$ if $r\rightarrow 0$, for the boundedness of the electric/magnetic field at the origin one selects only the Bessel function of the first kind J for the interior region. In the outer region, we are looking for a complex superposition of J and Y that represents outgoing waves. Such a solution can be given in terms of the Hankel functions of the first kind H$^{(1)}$ or of the second kind H$^{(2)}$. Using the asymptotic expansions of these functions [73, chap. 9, Eq. (9.2.3), Eq. (9.2.4)]

$$\begin{aligned}\mbox{H}^{(1)}_{\nu}(nkr) & \sim \sqrt{\frac{2}{... ...isplaystyle - \mbox{i}(nkr - \nu \pi /2 - \pi /4)}, \end{aligned}$$

and taking into account the harmonic time dependence $\exp($i$\omega t)$ (with positive frequency), one observes that H$^{(1)}$ represents incoming waves, while outgoing waves are given by H$^{(2)}$. Thus the piecewise ansatz for the basic components of the electric/magnetic bent mode profile is

$$\phi(r) = \left \{ \begin{array}{ll} A_{\mbox{\scriptsize s}}\mbo... ...(n_{\mbox{\scriptsize c}}k r), & \mbox{for~~} r \geq R^{+}, \end{array} \right.$$ (2.6)

where $R^{-} = R-d$, $R^{+} = R$, and where $A_{\mbox{\scriptsize s}}$, $A_{\mbox{\scriptsize f}}$, $B_{\mbox{\scriptsize f}}$ and $A_{\mbox{\scriptsize c}}$ are so far unknown constants.

The polarization dependent interface conditions lead to a homogeneous system of linear equations for $A_{\mbox{\scriptsize s}}$, $A_{\mbox{\scriptsize f}}$, $B_{\mbox{\scriptsize f}}$ and $A_{\mbox{\scriptsize c}}$. The condition for a nontrivial solution can be given the form

$$\frac{\displaystyle ~\frac{\mbox{J}_{\nu}(n_{\mbox{\scriptsize f}... ...tsize f}}k R^{+})}{\mbox{H}^{(2)^{'}}_{\nu}(n_{\mbox{\scriptsize c}}k R^{+})}~}$$ (2.7)

with $q_j = n_{\mbox{\scriptsize f}} / n_{j}$ for TE polarization, and with $q_j = n_{j} / n_{\mbox{\scriptsize f}}$ for TM polarized fields, for $j=$s$,$   c. Eq. (2.7) is the dispersion equation for the three layer bent slab waveguide. For given frequency $\omega$, this equation is to be solved^2.2 for the propagation constants $\gamma = \nu/R$.

For the numerical implementation, Eq. (2.7) is rearranged as

$$T_{1}\cdot T_{2} = T_{3} \cdot T_{4},$$ (2.8)

where

$$T_{1} = _{\nu}(n_{\mbox{\scriptsize f}} k R^{-}) \ \mbox{J}_{\nu}^{'}(n_{... ...scriptsize s}} k R^{-}) \ \mbox{J}_{\nu}^{'}(n_{\mbox{\scriptsize f}} k R^{-}),$$

$$T_{2} = _{\nu}(n_{\mbox{\scriptsize f}} k R^{+}) \ \mbox{H}^{(2)^{'}}_{\n... ...scriptsize c}} k R^{+}) \ \mbox{Y}_{\nu}^{'}(n_{\mbox{\scriptsize f}} k R^{+}),$$

$$T_{3} = _{\nu}(n_{\mbox{\scriptsize f}} k R^{-}) \ \mbox{J}_{\nu}^{'}(n_{... ...scriptsize s}} k R^{-}) \ \mbox{Y}_{\nu}^{'}(n_{\mbox{\scriptsize f}} k R^{-}),$$

$$T_{4} = _{\nu}(n_{\mbox{\scriptsize f}} k R^{+}) \ \mbox{H}^{(2)^{'}}_{\n... ...scriptsize c}} k R^{+}) \ \mbox{J}_{\nu}^{'}(n_{\mbox{\scriptsize f}} k R^{+}).$$

In contrast to common notions about leaky modes, the fields obtained by the ansatz (2.6) do not diverge for large radial coordinate $r$. The asymptotic expansion (2.5) predicts a decay $\sim 1/\sqrt{r}$. No difficulties related to `large' terms are to be expected for the numerical evaluation of Eq. (2.8). Moreover, as shown below, with the squared mode profile being accompanied by a factor $r^{-1}$ in the relevant expression, the bend modes can even be normalized with respect to the azimuthal mode power.