Orthogonality of bend modes

If the bend mode profiles are employed as basis elements for an expansion of a general optical field in the bend structure, the orthogonality properties of these modes become relevant: Projecting on the basis modes allows to relate the modal amplitudes to the given arbitrary field. As a consequence of the leaky nature of the complex bend modes, the orthogonality relations involve nonconjugate versions of the field profiles.^2.3

Let $(\boldsymbol {E}_p, \boldsymbol {H}_{p})$ and $(\boldsymbol {E}_q, \boldsymbol {H}_{q})$ be the electromagnetic fields (2.1) of bend modes with propagation constants $\gamma_{p}$ and $\gamma_{q}$, respectively, that are supported by the same bent waveguide. We start with the identity

$$\nabla \cdot (\boldsymbol {E}_{p} \times \boldsymbol {H}_{q} - \boldsymbol {E}_{q} \times \boldsymbol {H}_{p}) = 0,$$ (2.13)

which is a straightforward consequence of the Maxwell equations.

Consider the integral of Eq. (2.13) over an angular segment $\Omega = [0, \tilde{r}] \times [\theta, \theta + \Delta \theta]$ in the waveguide plane, specified by intervals of the polar coordinates:

$$\int_{\Omega} \nabla \cdot \boldsymbol {A} \, S= 0, \quad \boldsymbol {A} = (A_{r}, A_{y}, A_{\theta}) = \boldsymbol ... ...p} \times \boldsymbol {H}_{q} - \boldsymbol {E}_{q} \times \boldsymbol {H}_{p}.$$ (2.14)

After simplification of Eq. (2.14) by means of the Gauss theorem and a Taylor series expansion around $\theta$, for small, nonzero $\Delta \theta$ one obtains

$$(\gamma_{p}+\gamma_{q}) R \int_{0}^{\tilde{r}} A_{\theta}(r, \theta) \, r = \tilde{r} A_{r}(\tilde{r}, \theta).$$ (2.15)

In order to evaluate the limit $\tilde{r} \rightarrow \infty$ of the right hand side, $r A_r$ is expressed in terms of the basic mode profile components $E_y$ and $H_y$, with the help of Eqs. (2.2, 2.3):

$$r A_{r} = {\displaystyle \frac{\mbox{i}}{\mu_0 \omega }} r \left(... ...ega}} r \left( H_{p,y} \partial_r H_{q,y} - H_{q,y} \partial_r H_{p,y} \right).$$ (2.16)

Here $\epsilon_{\mbox{\scriptsize c}} = n_{\mbox{\scriptsize c}}^2$ is the permittivity in the exterior region of the bend (constant for large radii). In this region, the basic components $\phi_p = \tilde{E}_{p, y}$ or $\phi_p=\tilde{H}_{p, y}$ of modal solutions (2.6) are given by Hankel functions of second kind i.e. $\phi_p(r) = A_{\mbox{\scriptsize c},p}\,\mbox{H}^{(2)}_{\gamma_p R}(n_{\mbox{\scriptsize c}} k r)$, which, for large radial coordinate, assume the asymptotic forms

$$\begin{aligned}\phi_{p}(r) & \sim {\displaystyle A_{\mbox{\scri... ...box{i} n_{\mbox{\scriptsize c}} k\right)\phi_p(r)}. \end{aligned}$$

By using these expressions, the limits $r \rightarrow \infty$ of the individual parts of Eq. (2.16) can be shown to vanish

$$\lim_{r \rightarrow \infty} \left[ r(\phi_{p} \partial_r \phi_{q} - \phi_{q} \partial_r \phi_{p})\right] = 0.$$ (2.18)

This leads to the identity

$$(\gamma_{p} + \gamma_{q}) \int_{0}^{\infty} \boldsymbol {a}_{\the... ... \times \boldsymbol {H}_{q} - \boldsymbol {E}_{q} \times \boldsymbol {H}_{p})\, r= 0,$$ (2.19)

where $\boldsymbol {a}_{\theta}$ is the unit vector in the azimuthal ($\theta$-) direction.

After inspecting Eqs. (2.2), (2.3), one readily sees that the fields $(\boldsymbol {E}_{\tilde{p}}, \boldsymbol {H}_{\tilde{p}})$ and the propagation constant $\gamma_{\tilde{p}}$ with

$$\begin{aligned}\gamma_{\tilde{p}} &= - \gamma_{p}, \\ E_{\tilde... ...y} &= -H_{p,y}, H_{\tilde{p},\theta} = H_{p,\theta} \end{aligned}$$

describe a valid modal solution of the bend problem. By writing out the expression (2.19) for the quantities with indices $\tilde{p}$ and $q$, by applying the transformation (2.20), and by observing that $\boldsymbol {a}_{\theta} \cdot ( \boldsymbol {E}_{\tilde{p}} \times \boldsymbo... ...p} \times \boldsymbol {H}_{q} + \boldsymbol {E}_{q} \times \boldsymbol {H}_{p})$, Eq. (2.19) can be given the form

$$(\gamma_{p} - \gamma_{q}) \int_{0}^{\infty} \boldsymbol {a}_{\the... ... \times \boldsymbol {H}_{q} + \boldsymbol {E}_{q} \times \boldsymbol {H}_{p})\, r= 0.$$ (2.21)

Motivated by the result (2.21), we define the following symmetric, complex valued product^2.4 of two (integrable) electromagnetic fields $(\boldsymbol {E}_1, \boldsymbol {H}_1)$ and $(\boldsymbol {E}_2, \boldsymbol {H}_2)$, given in the polar coordinate system of the bend structure:

$$\hspace{-1.0cm} (\boldsymbol {E}_1, \boldsymbol {H}_1; \boldsymbol {E}_2, \boldsymbol {H}_2) = \hspace{-0.08cm} \int_0^\infty \boldsymbol {a}_{\theta}\cdot(\boldsymbol {E}_{1}\times\boldsymbol {H}_{2} + \boldsymbol {E}_{2} \times \boldsymbol {H}_{1})\, r$$

$$= \hspace{-0.2cm} \int_0^\infty \hspace{-0.2cm} ( E_{1,r}H_{2,y}\hs... ...pace{-0.1cm} E_{2,r}H_{1,y} \hspace{-0.1cm} - \hspace{-0.1cm} E_{2,y}H_{1,r})\, r.$$ (2.22)

Obviously, the integrand vanishes if fields of different (2-D) polarizations are inserted, i.e. TE and TM bend modes are orthogonal with respect to (2.22). One easily checks that the product is also zero, if the forward and backward versions (two fields with their components related by the transformation (2.20)) of a bend mode are inserted. Finally, according to Eq. (2.21), two nondegenerate bend modes with propagation constants $\gamma_p \neq \gamma_q$ that are supported by the same bend structure are orthogonal with respect to the product (2.22). These formal statements hold for pairs of the fields (2.1) with the full space and time dependence, for the expressions excluding the time dependence, as well as for pairs of pure mode profiles that depend on the radial coordinate only.

Assuming that for a given bend configuration a discrete, indexed set of nondegenerate modal fields $(\boldsymbol {E}_p, \boldsymbol {H}_p)$ with (pairwise different) propagation constants $\gamma_p$ is considered, the orthogonality properties can be stated in the more compact form

$$(\boldsymbol {E}_p, \boldsymbol {H}_p; \boldsymbol {E}_q, \boldsymbol {H}_q) = \delta_{p,q} N_{p},$$ (2.23)

with

$$N_{p} = 2 \int_{0}^{\infty} \boldsymbol {a}_{\theta} \cdot ( \boldsymbol {E}_{p} \times \boldsymbol {H}_{p} )\, r= 2 \int_0^\infty (E_{p,r}H_{p,y}-E_{p,y}H_{p,r})\, r,$$

and $\delta_{p,q} = 0$ for $p \neq q$, $\delta_{p,p} = 1$. For mode sets of uniform polarization and uniform direction of propagation, it can be convenient to write the orthogonality properties in terms of the basic mode profile components $\phi = \tilde{E}_y$ (TE) or $\phi = \tilde{H}_y$ (TM). This leads to the relations

$$\int_{0}^{\infty} \zeta \,\frac{ \phi_{p} \phi_{q}}{r} \, r= \delta_{p,q} P_p, P_{p} = \int_{0}^{\infty} \zeta\,\frac{\phi_{p}^2}{r}\, r,$$ (2.24)

$\zeta=1$ for TE, and $\zeta = 1/\epsilon(r)$ for TM polarization, which differ from the corresponding familiar expressions for straight dielectric slab waveguides by the appearance of the inverse radial coordinate $r$ only. According to Eq. (2.17), $P_p$ is obviously bounded. Note, however, that here $N_p$ and $P_p$ are complex valued quantities.

An alternative derivation of Eq. (2.24) starts with the eigenvalue equation (2.4), written out for two different modal solutions. Each equation is multiplied by the other mode profile, one subtracts the results, and integrates over the radial axis. This leads to an equation with the difference of the squared propagation constants times the integral of Eq. (2.24) on one side, and with a limit as in Eq. (2.16) on the other. Then the reasoning of Eqs. (2.17, 2.18) can be applied to obtain the desired result. All sets of bend mode profiles shown in the following sections satisfy the relations (2.23) or (2.24), respectively, up to the accuracy that can be expected from the computational procedures.