Derivation of perturbational expression

For a bent waveguide with the refractive index distribution $n(r)=\sqrt{\epsilon(r)}$, let $(\boldsymbol {E}, \boldsymbol {H})$ be the full electric and magnetic field (2.1) for a given mode. Suppose that the core refractive index is slightly perturbed, and the perturbed refractive index distribution is given by $n_{p}(r)=\sqrt{\epsilon_{p}(r)}$. For this perturbation, assuming that the mode profile remains unchanged, the corresponding perturbed mode $(\boldsymbol {E}_{p},\boldsymbol {H}_{p})$ is approximated as

$$\begin{pmatrix}\boldsymbol {E}_{p} \\ \boldsymbol {H}_{p} \end{pm... ...x} = P(\theta) \begin{pmatrix}\boldsymbol {E} \\ \boldsymbol {H} \end{pmatrix},$$ (2.26)

where $P(\theta)$ is an unknown function of the angular coordinate $\theta$.

By using Lorentz's reciprocity theorem [43] in polar coordinates to $(\boldsymbol {E}_{p}, \boldsymbol {H}_{p}, \epsilon_{p})$ and $(\boldsymbol {E}, \boldsymbol {H}, \epsilon)$, one obtains

$$\int_{0}^{\infty} \nabla \cdot (\boldsymbol {E}_{p} \times \boldsymbol {H}^{*} + \boldsymbol {E}^{*} \times \boldsymbol {H}_{p} ) \ r \ r= - \omega \epsilon_{0} \int_{0}^{\infty} (\epsilon_{p} - \epsilon) \boldsymbol {E}_{p} \cdot \boldsymbol {E}^{*} \ r \ r,$$

which upon inserting the ansatz given by Eq. (2.26) and after simplifying reduces to

$$\frac{d P}{d \hspace{0.05cm}\theta} \int_{0}^{\infty} \boldsymbol... ...E} \times \boldsymbol {H}^{*} + \boldsymbol {E}^{*} \times \boldsymbol {H} ) \ r= - \omega \epsilon_{0} P \int_{0}^{\infty} (\epsilon_{p} - \epsilon) \boldsymbol {E} \cdot \boldsymbol {E}^{*} \ r \ r,$$ (2.27)

where $\boldsymbol {a}_{\theta}$ is the unit vector in the angular direction.

Inserting the bent waveguide field ansatz given by Eq. (2.1) and solving for $P(\theta)$ leads to

$$P(\theta) = P_{0} \ \exp{\left ( - \mbox{i}\omega \epsilon_{0} \frac{\int_{0}... ...tilde{E}}^{*} \times \boldsymbol {\tilde{H}} ) \ \mbox{d}r} \ \theta \right) },$$ (2.28)

where $P_{0}$ is a constant, the superscript $\sim$ represents the mode profile associated with the field. Thus the perturbed modal field is

$$\begin{pmatrix}\boldsymbol {E}_{p} \\ \boldsymbol {H}_{p} \end{pm... ...E}}^{*} \times \boldsymbol {\tilde{H}} ) \ \mbox{d}r} \right)R \theta \right)},$$

and the change in propagation constant $\delta \gamma$ due to the perturbation is given by

$$\delta \gamma = \frac{\omega \epsilon_{0}}{R} \frac{\int_{0}^{\in... ...*} + \boldsymbol {\tilde{E}}^{*} \times \boldsymbol {\tilde{H}} ) \ \mbox{d}r}.$$ (2.29)

Note that above expression can also be written in terms of modal fields $(\boldsymbol {E}, \boldsymbol {H})$ instead of mode profiles $(\boldsymbol {\tilde{E}}, \boldsymbol {\tilde{H}})$. The right hand side of Eq. (2.29) is a pure real number. Therefore this expression, in fact, gives the change in the real part of the propagation constant only. In Ref. [43] a similar expression for the change in propagation constant for bulk uniform permittivity perturbations of straight waveguides is derived by means of a variational principle.

The use of asymptotic expansion of H$^{(2)}_{\nu}(nkr)$, given by Eq. (2.5), reveals that, in the present case of bent waveguides, the integral $\int_{0}^{\infty} (\epsilon_{p} - \epsilon) \boldsymbol {\tilde{E}} \cdot \boldsymbol {\tilde{E}}^{*} \ r \$   d$r$ is undefined for the upper limit $r=\infty$, if $\epsilon_{p} - \epsilon$ does not vanish for large radial coordinates. Still for a uniform perturbation $\delta \epsilon_{\mbox{\scriptsize f}} = \delta n^{2}_{\mbox{\scriptsize f}} = n_{\mbox{\scriptsize f},p}^{2} - n_{\mbox{\scriptsize f}}^{2}$ of the core refractive index, it is well defined; in that case Eq. (2.29) simplifies to

$$\delta \beta = \frac{\omega \epsilon_{0}}{R} \frac{\delta n^{2}_{... ...\boldsymbol {H}^{*} + \boldsymbol {E}^{*} \times \boldsymbol {H} ) \ \mbox{d}r}$$ (2.30)

where $R-d$ and $R$ define the core interface as shown in Figure 2.1, $n_{\mbox{\scriptsize f},p}$ and $n_{\mbox{\scriptsize f}}$ are perturbed and unperturbed core refractive index respectively. For a small uniform perturbation of the core refractive index, using Eq. (2.30), one can approximately write

$$\frac{\partial \beta}{\partial n_{\mbox{\scriptsize f}}} = 2 n_{\... ...boldsymbol {H}^{*} + \boldsymbol {E}^{*} \times \boldsymbol {H} ) \ \mbox{d}r}.$$ (2.31)

Note that the integrals that occur in the above expression are well behaved.