Spiral modes supported by circular dielectric tubes and tube segments

Abstract

The modal properties of curved dielectric slab waveguides are investigated. We consider quasi-confined, attenuated modes that propagate at oblique angles with respect to the axis through the center of curvature. Our analytical model describes the transition from scalar 2-D TE/TM bend modes to lossless spiral waves at near-axis propagation angles, with a continuum of vectorial attenuated spiral modes in between. Modal solutions are characterized in terms of directional wavenumbers and attenuation constants. Examples for vectorial mode profiles illustrate the effects of oblique wave propagation along the curved slab segments. For the regime of lossless spiral waves, the relation with the guided modes of corresponding dielectric tubes is demonstrated.

Appendix: Continuity of spiral mode profiles

We refer to the bend configuration as introduced in Fig. 1, and the formalism of Sects. 2.1–2.1.2. Principal field components $E_y$ and $H_y$ are defined piecewise for the regions $r<R-d$, $R-d< r< R$, and $R< r$ in the form of Eq. (8), with separate coefficients $A_E$, $B_E$, $C_E$, $D_E$, and $A_H$, $B_H$, $C_H$, $D_H$, respectively. Equations. (5) relate the principal fields to the remaining electromagnetic components $E_\theta$, $H_\theta$, $E_r$, and $H_r$.

Requiring the principal components $E_y$ and $H_y$, and the angular components $E_\theta$ and $H_\theta$, to be continuous at $r= R^- = R-d$ leads to the equations

$$\begin{aligned}&A_E{\mathrm {J}}_{\nu }(R^{-}\chi _{\mathrm{s}})= B_E{\mathrm {J}}_{\nu }(R^{-}\chi _{\mathrm{f}})+C_E{\mathrm {Y}}_{\nu }(R^{-}\chi _{\mathrm{f}}), \end{aligned}$$

(14)

$$\begin{aligned}&A_H{\mathrm {J}}_{\nu }(R^{-}\chi _{\mathrm{s}})=B_H{\mathrm {J}}_{\nu }(R^{-} \chi _{\mathrm{f}})+C_H{\mathrm {Y}}_{\nu }(R^{-}\chi _{\mathrm{f}}), \end{aligned}$$

(15)

$$\dfrac{1}{\chi _{\mathrm{s}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R^{-}}A_E{\mathrm {J}}_{\nu }(R^{-}\chi _{\mathrm{s}})-\omega \mu _{0}\chi _{\mathrm{s}} A_H{\mathrm {J'}}_{\nu }(R^{-}\chi _{\mathrm{s}})\right) = \dfrac{1}{\chi _{\mathrm{f}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R^{-}}\left( B_E{\mathrm {J_{\nu }}}(R^{-}\chi _{\mathrm{f}}) +C_E{\mathrm {Y}}_{\nu }(R^{-}\chi _{\mathrm{f}})\right) -\omega \mu _{0}\chi _{\mathrm{f}}\left( B_H{\mathrm {J'}}_{\nu }(R^{-}\chi _{\mathrm{f}}) +C_H{\mathrm {Y'}}_{\nu }(R^{-}\chi _{\mathrm{f}})\right) \right) ,$$

(16)

$$\begin{aligned}&\dfrac{1}{\chi _{\mathrm{s}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R^{-}}A_H{\mathrm {J}}_{\nu }(R^{-}\chi _{\mathrm{s}}) +\omega \epsilon _{0}n_{\mathrm{s}}^2\chi _{\mathrm{s}} A_E{\mathrm {J'}}_{\nu }(R^{-}\chi _{\mathrm{s}})\right) \nonumber \\&\quad = \dfrac{1}{\chi _{\mathrm{f}}^2}\left(-{\mathrm {i}}\dfrac{k_{y}\nu }{R^{-}}\left( B_H{\mathrm {J}}_{\nu }(R^{-}\chi _{\mathrm{f}}) +C_H{\mathrm {Y}}_{\nu }(R^{-}\chi _{\mathrm{f}})\right) +\omega \epsilon _{0}n_{\mathrm{f}}^2\chi _f \left( B_E{\mathrm {J'}}_{\nu }(R^{-} \chi _{\mathrm{f}})+C_E{\mathrm {Y'}}_{\nu }(R^{-}\chi _{\mathrm{f}})\right) \right). \end{aligned}$$

(17)

Likewise, these four fields are continuous at the outer interface at $r=R$, if the equations

$$\begin{aligned}&B_E{\mathrm {J}}_{\nu }(R\chi _{\mathrm{f}})+C_E{\mathrm {Y}}_{\nu }(R\chi _{\mathrm{f}})=D_E{\mathrm {H}}_{\nu }^{(2)}(\pm R\chi _{\mathrm{c}}), \end{aligned}$$

(18)

$$\begin{aligned}&B_H{\mathrm {J}}_{\nu }(R\chi _{\mathrm{f}})+C_H{\mathrm {Y}}_{\nu }(R\chi _{\mathrm{f}})=D_H{\mathrm {H}}_{\nu }^{(2)}(\pm R\chi _{\mathrm{c}}), \end{aligned}$$

(19)

$$\begin{aligned}&\dfrac{1}{\chi _{\mathrm{f}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R}\left( B_E{\mathrm {J}}_{\nu }(R\chi _{\mathrm{f}}) +C_E{\mathrm {Y}}_{\nu }(R\chi _{\mathrm{f}})\right) -\omega \mu _{0}\chi _{\mathrm{f}}\left( B_H{\mathrm {J'}}_{\nu }(R\chi _{\mathrm{f}})+C_H{\mathrm {Y'}}_{\nu }(R\chi _{\mathrm{f}})\right) \right) \nonumber \\&\quad = \dfrac{1}{\chi _{\mathrm{c}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R}D_E{\mathrm {H}}^{(2)}_{\nu }(\pm R\chi _{\mathrm{c}}){\mp }\omega \mu _{0}\chi _{\mathrm{c}}D_H{\mathrm {H}}_{\nu }^{(2)'}(\pm R\chi _{\mathrm{c}})\right) , \end{aligned}$$

(20)

$$\begin{aligned}&\dfrac{1}{\chi _{\mathrm{f}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R}\left( B_H{\mathrm {J}}_{\nu }(R\chi _{\mathrm{f}})+C_H{\mathrm {Y}}_{\nu }(R\chi _{\mathrm{f}})\right) +\omega \epsilon _{0}n_{\mathrm{f}}^2\chi _{\mathrm{f}}\left( B_E{\mathrm {J'}}_{\nu }(R\chi _{\mathrm{f}})+C_E{\mathrm {Y'}}_{\nu }(R\chi _{\mathrm{f}})\right) \right) \nonumber \\&\quad = \dfrac{1}{\chi _{\mathrm{c}}^2}\left( -{\mathrm {i}}\dfrac{k_{y}\nu }{R}D_H{\mathrm {H}}^{(2)}_{\nu }(\pm R\chi _{\mathrm{c}})\pm \omega \epsilon _{0}n_{\mathrm{c}}^2\chi _{\mathrm{c}}D_E{\mathrm {H}}^{(2)'}_{\nu }(\pm R\chi _{\mathrm{c}})\right) \end{aligned}$$

(21)

are satisfied. Here abbreviations $\chi _r=\sqrt{k_0^2n_r^2-k_y^2}$, for $r \in \{{\mathrm {s}},{\mathrm {f}},{\mathrm {c}}\}$, have been introduced. Signs ± and ${\mp }$ distinguish wavenumber parameters $k_y\,\lessgtr\,k_0n_{\mathrm{c}}$. Just as in the discussion of Eq. (8), the √-symbol is meant to indicate the positive real root, for a positive radicand, or the imaginary root with positive imaginary part, in case of a negative radicand. Eqs. (14)–(21) imply continuity of $H_r$ and of $n^2 E_r$ at the radial positions of both interfaces.