Bend mode normalization

The power flow density associated with a bend mode is given by the time averaged Poynting vector $\boldsymbol {S}_{\mbox{\scriptsize av}}= \frac{1}{2} \Re{(\boldsymbol {E} \times \boldsymbol {H}^{*})}$. The axial component $S_{\mbox{\scriptsize av}, y}$ vanishes in the 2-D setting; for TE waves the radial and azimuthal components evaluate to

$$\begin{aligned}S_{\mbox{\scriptsize av},r} & = \frac{ -1 }{2 \m... ...rt^{2} \mbox{e}^{\displaystyle -2 \alpha R \theta}, \end{aligned}$$

and for TM polarization one obtains

$$\begin{aligned}S_{\mbox{\scriptsize av},r} & = \frac{ 1 }{2 \ep... ...vert^2 \mbox{e}^{\displaystyle -2 \alpha R \theta}. \end{aligned}$$

The total optical power transported by the mode in the angular direction is given by $P_{\theta} (\theta) = \int_{0}^{\infty} S_{\mbox{\scriptsize av},\theta}\,\mbox{d}r$. Somewhat surprisingly, this expression can be considerably simplified by using the following formula [89, Section 11.2, Eq. 5],

$$\int C_{\mu} ( kx ) D_{\nu} ( kx ) \frac{\mbox{d}x}{x} = \frac{kx}{ \mu^{2} - \nu^{2}} \left\{ C_{\mu}(kx) D_{\nu + 1}(kx) - C_{\mu + 1}(kx) D_{\nu}(kx) \right\}$$

$$+ \frac{ C_{\mu}(kx) D_{\nu}(kx)}{ \mu + \nu}$$ (2.11)

where $C_{\mu}$, $D_{\nu}$ are any cylindrical functions (i.e. functions which are linear combinations of J$_\mu$ and Y$_\mu$, or of J$_\nu$ and Y$_\nu$, respectively). Observing that for a valid mode profile the pieces of the ansatz (2.6) satisfy the polarization dependent continuity conditions at the dielectric interfaces, application of Eq. (2.11) and of several standard identities for Bessel functions leads to exact cancellation of the boundary terms that arise in the piecewise integration, with the exception of the limit term for $r \rightarrow \infty$. In that regime the mode profile is represented by the asymptotic form (2.5) of the relevant Hankel functions, such that one arrives at the two expressions

$$\begin{aligned}P_{\theta} (\theta) & = \displaystyle{\frac{\ver... ...ystyle \alpha R ( \pi - 2 \theta )}} \mbox{~~(TM),} \end{aligned}$$

for the modal power of TE and TM polarized modes, respectively. For certain well guided modes with extremely low losses, i.e. $\alpha \approx 0$, Eqs. (2.12) are not suitable for direct use. In this case, we compute the modal power by numerical integration of $P_{\theta} (\theta) = \int_{0}^{\infty} S_{\mbox{\scriptsize av},\theta}\,\mbox{d}r$ over a suitably chosen radial interval. All mode profiles shown in Section 2.4 are power normalized with respect to these expressions (evaluated at $\theta=0$).

Alternatively, Eqs. (2.12) can be derived in a way quite analogous to what follows in Section 2.2.2: Upon integrating the vanishing divergence of the Poynting vector $\nabla\cdot(\boldsymbol {E} \times \boldsymbol {H}^\ast + \boldsymbol {E}^\ast \times\boldsymbol {H}) = 0$ for a modal solution $(\boldsymbol {E}, \boldsymbol {H})$ over a differential angular segment in the domain of polar coordinates, by means of Gauss' theorem one relates the angular decay of modal power to the outflow of optical power in the radial direction. The limit of that flow for large radial coordinates exists and can be evaluated by again using the asymptotic form (2.5) of the mode profile, leading to expressions (2.12) for the modal power.

By considering the above expressions for large bend radii, one might wonder whether these may lead to a scheme for the normalization of nonguided modal solutions associated with straight waveguides e.g. as given in [90] in terms of plane wave superpositions. Examination of Eq. (2.12) with the help of Eq. (2.25), however shows that the expression (2.12) for the modal power is not applicable in the limit $R \rightarrow \infty$. Hence in this respect, there is no direct correspondence between the present bend modes supported by structures with low curvature and radiative modes of similar straight waveguides.