Coupled-mode analysis for chiral fiber gratings with a core enclosed by a non-circular equiwidth curve

Abstract

Polarization properties of chiral fiber gratings with the core enclosed by a non-circular equiwidth curve have been analyzed and examined numerically. Since the curve has multiple odd number degree of symmetries, the specific analysis is mainly based on the simplest triple-helix chiral fiber gratings and difference of properties caused by small geometric deformation. Results show that, only right-handed circularly polarized core modes couples with cladding modes. Besides, if the shape of the fiber core was closer to a circle, rather than a triangle with curved edges, both the shortest total power transfer length and the distance between the relevant resonant dips of the transmission spectrum would increase. Therefore, as a multiple-helix chiral fiber grating, it might have some potential applications, such as filters, polarizers, and sensors.

Appendix A

In this appendix the fields of the core and the cladding modes in a step index fiber are summarized. An approximate model for cladding modes can be applied to suggest an analytical expression in the theoretical analysis in the limit of a wide-cladding radius (R_cl >> λ) and small index contrast (n_co − n_cl << n_cl). With the initial small core neglected, the analytical expressions of the core and the cladding modes are in the same form, except that index profiles for the core and the cladding modes are different. Since the eigenmodes of the HE_l+1,m and EH_l−1,m are degenerated, only expressions for the HE modes are listed as follows:

$$\left\{ \begin{gathered} e_{rm} (r,\theta ) = F_m (r)\left\{ {\left. \begin{gathered} \cos (m\theta ) \hfill \\ \sin (m\theta ) \hfill \\ \end{gathered} \right\}} \right. \hfill \\ e_{\theta m} (r,\theta ) = F_m (r)\left\{ {\left. \begin{gathered} - \sin (m\theta ) \hfill \\ \cos (m\theta ) \hfill \\ \end{gathered} \right\}} \right. \hfill \\ e_{zm} (r,\theta ) = E_{zm} (r)\left\{ {\left. \begin{gathered} \cos (m\theta ) \hfill \\ \sin (m\theta ) \hfill \\ \end{gathered} \right\}} \right. \hfill \\ \end{gathered} \right.$$

(A1)

In Eqs. (A1), the upper and lower expressions in the same curly bracket indicate the even and odd modes, respectively.

In the above equations,

$$F_m (r) = E_0 \left\{ {\begin{array}{*{20}l} { - i\frac{a\beta_m }{{UJ_m (U)}}J_{m - 1} \left( {U\frac{r}{a}} \right)} \hfill & {0 \le r \le a} \hfill \\ { - i\frac{a\beta_m }{{WK_m (W)}}K_{m - 1} \left( {W\frac{r}{a}} \right)} \hfill & {a \le r < \infty } \hfill \\ \end{array} } \right.$$

(A2)

$$E_{zm} (r) = E_0 \left\{ {\begin{array}{*{20}l} {\frac{{J_m \left( {U\frac{r}{a}} \right)}}{J_m (U)}} \hfill & {0 \le r \le a} \hfill \\ {\frac{{K_m \left( {W\frac{r}{a}} \right)}}{K_m (W)}} \hfill & {a \le r < \infty } \hfill \\ \end{array} } \right.$$

(A3)

Then we define ${\text{HE}}_{lm}^+$ and ${\text{HE}}_{lm}^-$ as

$$\left\{ \begin{gathered} HE_{lm}^+ = (HE_{lm}^{even} ) + i(HE_{lm}^{odd} ) \hfill \\ HE_{lm}^- = (HE_{lm}^{even} ) - i(HE_{lm}^{odd} ) \hfill \\ \end{gathered} \right.$$

(A4)

where the superscripts + or − indicate the modes with positive or negative angular number as shown in Eq. (A5), and modes with the superscripts even or odd are the even or odd modes in Eqs. (A1). Substituting Eqs. (A1) into Eq. (A4), the mode expressions in Eq. (A4) become

$$\left\{ \begin{gathered} HE_{lm}^+ = [F_m (r)(\overrightarrow r + i\overrightarrow \theta ) + E_{zm} (r)\overrightarrow z ]e^{im\theta } \hfill \\ HE_{lm}^- = [F_m (r)(\overrightarrow r - i\overrightarrow \theta ) + E_{zm} (r)\overrightarrow z ]e^{ - im\theta } \hfill \\ \end{gathered} \right.$$

(A5)

We will use the mode expression defined in Eq. (A5) for coupled-mode analysis. In the zero-order approximation, ${\text{HE}}_{lm}^+$ is left-handed circularly polarized modes, and ${\text{HE}}_{lm}^-$ is right-handed circularly polarized modes.