Design and Mathematical Analysis of Chiral Optical Fibers for Generating Spin and Orbital Angular Momentum

Abstract. This study presents a detailed theoretical analysis of chiral fibers, focusing on their unique helical core structure and its implications for mode coupling. Chiral fibers are distinguished by their twisted cores, which follow a helical path inside the cladding. The research applies mode-coupling theory under weak guidance to analyze the interactions between linearly polarized modes (LP modes) in the two-mode regime. By introducing periodic perturbations, the study demonstrates efficient power exchange and phase matching between modes, highlighting the coupling mechanisms facilitated by the fiber's chirality. Key findings include the derivation of coupled-mode equations and their solutions, which describe the behavior of the transverse electric field components and the influence of refractive index variations. The analysis further reveals the generation of paired higher-order modes carrying spin angular momentum (SAM) and orbital angular momentum (OAM), with specific phase differences of π/2. The SAM arises from the orthogonality of the electric field components, while the OAM is attributed to the azimuthal phase dependence of the light. Additionally, the study discusses the conversion of the fundamental HE₁₁ mode into combinations of TE₀₁ and TM₀₁ modes under circularly polarized excitation with opposite handedness to the fiber helix. These results emphasize the potential of chiral fibers in enabling advanced photonic applications, such as mode manipulation and angular momentum encoding in optical systems.

Theoretical Analysis

A chiral fiber is a spun eccentric core fiber, in which the twist pitches are less than 1 mm. Since in a fiber there is always an eccentricity between core and cladding, then when the fiber is twisted, its core follows a helical path inside the cladding. A chiral fiber is almost a standard fiber except that its core follows a helical path in the cladding. Thus, this helical core can be considered as a deformation of a straight round core and it is possible to apply the theory of mode-coupling in the presence of helical core to the study of modes in twisted fibers.

Assuming weak-guidance, we use linearly polarized modes ($\mathrm{LP}$). In the so-called two-mode regime, there are actually six linearly polarized modes that may exist in the fiber. By using the conventional notation, we designate these modes as ${\mathrm{LP}}_{01}^x$, ${\mathrm{LP}}_{01}^y$, ${\mathrm{LP}}_{11}^{\mathrm{xe}}$, ${\mathrm{LP}}_{11}^{\mathrm{xo}}$, ${\mathrm{LP}}_{11}^{\mathrm{ye}}$, ${\mathrm{LP}}_{11}^{\mathrm{yo}}$. Namely according to the distribution of its electromagnetic fields, the ${\mathrm{LP}}_{11}$ mode has four forms: x- and y-polarized fields with azimuthal variation of $\cos \phi$ (two even modes) and $\sin \phi$ (two odd modes) respectively.

In an straight ideal optical fiber, the ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes do not couple and thus not exchange power with each other. In order to provide the coupling between these two sets of modes, one may introduce an index perturbation along the fiber by periodic perturbation. The periodic perturbation causes the coupling between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes and at the same time facilitates phase-matching between the two sets of modes so that efficient power exchange between them may be achieved. In a chiral fiber, if the offset of the eccentric core is very small compared with the core diameter, the perturbation of the dielectric constant in the core region can be expressed as:

$$\mathrm{\Delta }𝜀(r,\phi )=(n_{\mathrm{co}}^2-n_{\mathrm{cl}}^2)\delta (r-r_0)d\cos (\phi -\tau z)$$

(1)

where the twist rate $\tau =2\pi ∕p$ ( $p$ is the twist pitch), $n_{\mathrm{co}}$ and $n_{\mathrm{cl}}$ are refractive indices of the core and the cladding, respectively, $z$ is the longitudinal coordinate variable of helical-core fiber, $d$ is the offset, and $r_0$ is the equivalent radius, which is the zero order coefficient of the trigonometrical series of the eccentric core with a diameter of $a$ and can be evaluated, numerically, solving:

$$r_{\mathrm{co}}(\phi ,z)=r_0+d\cos (\phi -\tau z)$$

(2)

where $r_{\mathrm{co}}$ is the z-dependent distance of the core–cladding boundary from the center of the fiber. Eq. (2) is the lowest-order expansion (i.e. considering only the first coefficient $r_1=d$ ) of the core boundary profile $r_{\mathrm{co}}(\phi ,z)$ for the chiral fiber expanded in harmonic functions:

$$r_{\mathrm{co}}(\phi ,z)=r_0+\sum _n{r}_n\cos (n\phi -n\tau z)$$

(3)

The twist rate $\tau$ is assumed to be positive or negative for right- or left-handed helical structures, respectively.

Assume that the transverse electric field components of the perturbed fiber can be represented by a linear superposition of the $\mathrm{LP}$ modes of the unperturbed fiber

$$E_t=\sum _q{a}_q(z)\exp (-j{\beta }_{q}z)e_q(r,\phi )=\sum _q{A}_q(z)e_q(r,\phi )$$

(4)

where the subscript $q$ stands for the mode index of the $\mathrm{LP}$ modes and ${\beta }_q$, $e_q(r,\phi )$ are the propagation constants and the field patterns of the corresponding $\mathrm{LP}$ modes. $A_q(z)$ are the mode complex amplitude of $\mathrm{LP}$ modes. The envelopes of the mode amplitudes $a_q$ are $z$ dependent as a result of the coupling among the modes. Note that the slowly varying in $z$ expansion coefficients $a_q(z)$ are related to the rapidly varying expansion coefficients $A_q(z)$ by

$$A_q(z)=a_q(z)\exp (-j{\beta }_{q}z)$$

(5)

The coupled-mode equations for $a_q$ are derived by using (4) as a trial solution to the vector wave equation for the transverse electric fields of the perturbed fiber

$$\{{\nabla }^2+n^2(r,\phi ,z)k^2\}{E}_t=-{\nabla }_t(E_t\cdot {\nabla }_t\ln n^2)$$

(6)

and by making use of the scalar wave equations for the $\mathrm{LP}$ modes of the unperturbed fiber

$$\left\{{\nabla }_t^2+{\tilde{n}}^2(r)k^2-{\beta }_q^2\right\}{e}_q=0$$

(7)

where $\tilde{n}$ is the refractive index of the unperturbed fiber.

In the two-mode regime, considering only the first six linear modes, Eq. (4) reduces to:

$$E_t=\sum _{l=[0,1],\mu =[x,y],\nu =[e,o]}{A}_{l1}^{\mathrm{\mu \nu }}(z)e_{l1}^{\mathrm{\mu \nu }}(r,\phi )$$

(8)

where:

$$\begin{array}{rcll}{e}_{l1}^{\mathrm{\mu \nu }}(r,\phi )& =& F_{l1}(r)\left\{\begin{array}{cc}\hfill \cos l\phi \widehat{\mu }\hfill & \hfill \nu =\mathrm{even}\hfill \\ \hfill \sin l\phi \widehat{\mu }\hfill & \hfill \nu =\mathrm{odd}\hfill \end{array}\right\}& \text{(9)}\end{array}$$

Due to periodic perturbation (1) applied along the fiber, the ${\mathrm{LP}}_{01}^x$ mode may couple to the ${\mathrm{LP}}_{11}^{\mathrm{xe}}$ and ${\mathrm{LP}}_{11}^{\mathrm{xo}}$ while the coupling to ${\mathrm{LP}}_{11}^{\mathrm{ye}}$ and ${\mathrm{LP}}_{11}^{\mathrm{yo}}$ modes which have different polarization will not occur because the index perturbation is isotropic. In the same manner the ${\mathrm{LP}}_{01}^y$ mode may couple to the ${\mathrm{LP}}_{11}^{\mathrm{ye}}$ and ${\mathrm{LP}}_{11}^{\mathrm{yo}}$. Therefore the pair of ${\mathrm{LP}}_{01}$ modes couple with both the two pairs of even and odd modes and a separate treatment of coupled equations for even and odd modes is valid because there are no couplings between the even and the odd modes. According to conventional coupled mode theory, the coupling between the ${\mathrm{LP}}_{01}$ modes and the even ${\mathrm{LP}}_{11}$ modes can be described by the following coupled-mode equations:

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}^x\hfill \\ \hfill A_{01}^y\hfill \\ \hfill A_{11}^{\mathrm{xe}}\hfill \\ \hfill A_{11}^{\mathrm{ye}}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill -j{\beta }_{01}\hfill & \hfill 0\hfill & \hfill -j{C}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -j{\beta }_{01}\hfill & \hfill 0\hfill & \hfill -j{C}_2\hfill \\ \hfill -j{C}_1\hfill & \hfill 0\hfill & \hfill -j{\beta }_{11}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -j{C}_2\hfill & \hfill 0\hfill & \hfill -j{\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}^x\hfill \\ \hfill A_{01}^y\hfill \\ \hfill A_{11}^{\mathrm{xe}}\hfill \\ \hfill A_{11}^{\mathrm{ye}}\hfill \end{array}\right]$$

(10)

The coupling coefficients $C_1$ and $C_2$ can be expressed as:

$$C_1=\omega {𝜖}_0\iint \mathrm{\Delta }𝜀(r,\phi )e_{01}^x\cdot e_{11}^{\mathrm{xe}}\mathrm{dS}=C_2=\omega {𝜖}_0\iint \mathrm{\Delta }𝜀(r,\phi )e_{01}^y\cdot e_{11}^{\mathrm{ye}}\mathrm{dS}$$

(11)

Substituting Eq. (1) to Eq. (11) , we have

$$C_1=C_2=C\cos (\tau z)$$

(12)

where

$$C=\frac{\mathrm{\pi \omega }{𝜖}_0(n_{\mathrm{co}}^2-n_{\mathrm{cl}}^2)r_{0}d{F}_{10}(r_0)F_{11}(r_0)}{4}$$

(13)

Then, the coupled-mode equations for the pair of ${\mathrm{LP}}_{01}$ modes and the pair of the even ${\mathrm{LP}}_{11}$ modes is rewritten as

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}^x\hfill \\ \hfill A_{01}^y\hfill \\ \hfill A_{11}^{\mathrm{xe}}\hfill \\ \hfill A_{11}^{\mathrm{ye}}\hfill \end{array}\right]=-j\left[\begin{array}{cccc}\hfill {\beta }_{01}\hfill & \hfill 0\hfill & \hfill C\cos (\tau z)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_{01}\hfill & \hfill 0\hfill & \hfill C\cos (\tau z)\hfill \\ \hfill C\cos (\tau z)\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill C\cos (\tau z)\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}^x\hfill \\ \hfill A_{01}^y\hfill \\ \hfill A_{11}^{\mathrm{xe}}\hfill \\ \hfill A_{11}^{\mathrm{ye}}\hfill \end{array}\right]$$

(14)

The coupled-mode equations for the pair of ${\mathrm{LP}}_{01}$ modes and the pair of the odd ${\mathrm{LP}}_{11}$ have the same form as Eq. (8), except that

$$C_1=\omega {𝜖}_0\iint \mathrm{\Delta }𝜀(r,\phi )e_{01}^x\cdot e_{11}^{\mathrm{xo}}\mathrm{dS}=C_2=\omega {𝜖}_0\iint \mathrm{\Delta }𝜀(r,\phi )e_{01}^y\cdot e_{11}^{\mathrm{yo}}\mathrm{dS}=-C\sin (\tau z)$$

(15)

Therefore for odd modes we have

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}^x\hfill \\ \hfill A_{01}^y\hfill \\ \hfill A_{11}^{\mathrm{xo}}\hfill \\ \hfill A_{11}^{\mathrm{yo}}\hfill \end{array}\right]=-j\left[\begin{array}{cccc}\hfill {\beta }_{01}\hfill & \hfill 0\hfill & \hfill -C\sin (\tau z)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_{01}\hfill & \hfill 0\hfill & \hfill -C\sin (\tau z)\hfill \\ \hfill -C\sin (\tau z)\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -C\sin (\tau z)\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}^x\hfill \\ \hfill A_{01}^y\hfill \\ \hfill A_{11}^{\mathrm{xe}}\hfill \\ \hfill A_{11}^{\mathrm{ye}}\hfill \end{array}\right]$$

(16)

From these two sets of coupled-mode equations, i.e, Eq. (14) and Eq. (16), x- and y-polarized modes are not coupled with each other. Thus, for both even and odd modes, we have two independent equation groups for the x- and y-polarized modes, respectively. The two equations groups for different polarizations also have the same forms. This means that light passing through the helical core fiber is polarization-independent. Then for an arbitrary polarization we have

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^e\hfill \end{array}\right]=-j\left[\begin{array}{cc}\hfill {\beta }_{01}\hfill & \hfill C\cos (\tau z)\hfill \\ \hfill C\cos (\tau z)\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^e\hfill \end{array}\right]$$

(17)

for the even ${\mathrm{LP}}_{11}$ mode, and

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^o\hfill \end{array}\right]=-j\left[\begin{array}{cc}\hfill {\beta }_{01}\hfill & \hfill -C\sin (\tau z)\hfill \\ \hfill -C\sin (\tau z)\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^e\hfill \end{array}\right]$$

(18)

for the odd ${\mathrm{LP}}_{11}$ mode.

Combining Eqs. (17) and (18) for an arbitrary polarization, we obtained the coupled-mode equation group, describing the coupling between the ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes with the same polarization state in a single-helix structure as

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^e\hfill \\ \hfill A_{11}^o\hfill \end{array}\right]=-j\left[\begin{array}{ccc}\hfill {\beta }_{01}\hfill & \hfill C∕2\cos (\tau z)\hfill & \hfill -C∕2\sin (\tau z)\hfill \\ \hfill C∕2\cos (\tau z)\hfill & \hfill {\beta }_{11}\hfill & \hfill 0\hfill \\ \hfill -C∕2\sin (\tau z)\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^e\hfill \\ \hfill A_{11}^o\hfill \end{array}\right]$$

(19)

Using a transformation of

$$A_{11}^{+}=\frac{A_{11}^e+j{A}_{11}^o}{\sqrt{2}}$$

(20)

and

$$A_{11}^{-}=\frac{A_{11}^e-j{A}_{11}^o}{\sqrt{2}}$$

(21)

Eq. (19) becomes

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^{+}\hfill \\ \hfill A_{11}^{-}\hfill \end{array}\right]=-j\left[\begin{array}{ccc}\hfill {\beta }_{01}\hfill & \hfill C∕\sqrt{2}\exp (\tau z)\hfill & \hfill C∕\sqrt{2}\exp (-j\tau z)\hfill \\ \hfill C∕\sqrt{2}\exp (-j\tau z)\hfill & \hfill {\beta }_{11}\hfill & \hfill 0\hfill \\ \hfill C∕\sqrt{2}\exp (j\tau z)\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^{+}\hfill \\ \hfill A_{11}^{-}\hfill \end{array}\right]$$

(22)

Pointing out that in the expression of the coefficients $A_q(z)$ given by (5), appears the exponential term $\exp (-j{\beta }_{q}z)$, it follows that in the coupled-mode equation (22) will attend terms with exponential dependence of the type $\exp (j({\beta }_{01}-{\beta }_{11}+\tau )z)$ (which multiplies $A_{11}^{+}$) or $\exp (j({\beta }_{01}-{\beta }_{11}-\tau )z)$ (which multiplies $A_{11}^{-}$) in correspondence of the matrix elements containing $\exp (j\tau z)$ or $\exp (-j\tau z)$ respectively.

For right-handed structure ( $\tau >0$) the phase matching condition is written as

$${\beta }_{01}-{\beta }_{11}-\tau =0$$

(23)

and, therefore, the $\exp (j({\beta }_{01}-{\beta }_{11}-\tau )z)$ results in a synchronous driving term (i.e., one with a zero exponent) while the contribution of $\exp (j({\beta }_{01}-{\beta }_{11}+\tau )z)$ is negligible.

Conversely, for left-handed structure ( $\tau <0$) the phase matching condition is written as

$${\beta }_{01}-{\beta }_{11}+\tau =0$$

(24)

and, therefore, only the $\exp (j({\beta }_{01}-{\beta }_{11}+\tau )z)$ is synchronous while $\exp (j({\beta }_{01}-{\beta }_{11}-\tau )z)$ is negligible.

Hence under phase matching condition, keeping only the synchronous term, the Eq. (22) can be further reduced to a 2 × 2 matrix equation expressed as

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^{-}\hfill \end{array}\right]=-j\left[\begin{array}{cc}\hfill {\beta }_{01}\hfill & \hfill C∕\sqrt{2}\exp (-j\tau z)\hfill \\ \hfill C∕\sqrt{2}\exp (j\tau z)\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^{-}\hfill \end{array}\right]$$

(25)

for a right-handed structure, and as

$$\frac{d}{\mathrm{dz}}\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^{+}\hfill \end{array}\right]=-j\left[\begin{array}{cc}\hfill {\beta }_{01}\hfill & \hfill C∕\sqrt{2}\exp (j\tau z)\hfill \\ \hfill C∕\sqrt{2}\exp (-j\tau z)\hfill & \hfill {\beta }_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A_{01}\hfill \\ \hfill A_{11}^{+}\hfill \end{array}\right]$$

(26)

for left-handed structure. This implies that different forms of ${\mathrm{LP}}_{11}$ modes are coupled out for different handed structure.

When the initial conditions are $A_{01}=1$ and $A_{11}^{+}=A_{11}^{-}=0$ at the input end ( $z=0$), Eq. (25) and (26) can be solved when the phase matching condition (23) and (24) are satisfied, and since $\tau$ is equal in absolute value in the two systems of differential equations (25) and (26) but has opposite sign, the two systems are formally identical, and then

$$A_{01}(z)=\cos \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{01}z)$$

(27)

$$A_{11}^{+}(z)=A_{11}^{-}(z)=-j\sin \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{11}z)$$

(28)

recalling the transformations (20) and (21), we obtain

$$\mathrm{for}\mathrm{right}-\mathrm{handed}\mathrm{helix}:\{\begin{array}{c}\hfill A_{11}^e(z)=-j\frac{\sqrt{2}}{2}\sin \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{11}z)\hfill \\ \hfill A_{11}^o(z)=-\frac{\sqrt{2}}{2}\sin \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{11}z)\hfill \end{array}$$

(29)

$$\mathrm{for}\mathrm{left}-\mathrm{handed}\mathrm{helix}:\{\begin{array}{c}\hfill A_{11}^e(z)=-j\frac{\sqrt{2}}{2}\sin \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{11}z)\hfill \\ \hfill A_{11}^o(z)=\frac{\sqrt{2}}{2}\sin \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{11}z)\hfill \end{array}$$

(30)

It follows

$$\{\begin{array}{cc}{A}_{11}^o(z)=-j{A}_{11}^e(z)\hfill & \mathrm{for}\mathrm{right}-\mathrm{handed}\mathrm{helix}\hfill \\ A_{11}^o(z)=j{A}_{11}^e(z)\hfill & \mathrm{for}\mathrm{left}-\mathrm{handed}\mathrm{helix}\hfill \end{array}$$

(31)

The fixed phase relationship in (31) imply that the superpositions of ${\mathrm{LP}}_{11}^o$ and ${\mathrm{LP}}_{11}^e$ modes are eigenmodes of the perturbed fiber.

Defining

$$A_{11}(z):=-j\sin \left(\frac{C}{\sqrt{2}}z\right)\exp (-j{\beta }_{11}z)$$

(32)

we can rewritten Eq. (29) and (30) as

$$\{\begin{array}{cc}{A}_{11}^e(z)=A_{11}(z),A_{11}^o(z)=-j{A}_{11}(z)\hfill & \mathrm{for}\mathrm{right}-\mathrm{handed}\mathrm{helix}\hfill \\ A_{11}^e(z)=A_{11}(z),A_{11}^o(z)=j{A}_{11}(z)\hfill & \mathrm{for}\mathrm{left}-\mathrm{handed}\mathrm{helix}\hfill \end{array}$$

(33)

By considering the interaction of a right (left) circularly polarizes $\mathrm{LP}{P}_{01}$ mode field $e_{01}^{r,l}$ and similarly polarized ${\mathrm{LP}}_{11}$ mode fields $e_{11}^{\mathrm{er},l}$ and $e_{11}^{\mathrm{or},l}$, we can rewrite (8) obtaining the transverse electric field

$$E_t^{r,l}=A_{01}(z)e_{01}^{r,l}(r,\phi )+A_{11}^{er,l}(z)e_{11}^{er,l}(r,\phi )+A_{11}^{or,l}(z)e_{11}^{or,l}(r,\phi )$$

(34)

for the polarization independence of the single helix chiral fiber we have $A_{11}^{er,l}=A_{11}^e$ and $A_{11}^{or,l}=A_{11}^o$

Substituting (33) in (34) we obtain

$$E_t^{r,l}=A_{01}(z)e_{01}^{r,l}(r,\phi )+A_{11}(z)e_{11}^{er,l}(r,\phi )\mp j{A}_{11}(z)e_{11}^{or,l}(r,\phi )$$

(35)

where the $-$ and $+$ in Eq. (35) correspond to right- and left-handed structure respectively.

Observing that $A_{01}(z)$ and $A_{11}(z)$ vary as a function respectively of the sine and cosine of $z$, a total power transfer periodically occurs between the fundamental mode ${\mathrm{LP}}_{01}=A_{01}(z)e_{01}^{r,l}(r,\phi )$ and the ${\mathrm{LP}}_{11}=A_{11}(z)e_{11}^{er,l}(r,\phi )\mp j{A}_{11}(z)e_{11}^{or,l}(r,\phi )$ mode. The optical power in the modes is, indeed, given by

$$\begin{array}{rcll}{\left|A_{01}\right|}^2& =& {\cos }^2\left(\frac{C}{\sqrt{2}}z\right)& \text{(36)}\\ {\left|A_{11}^e\right|}^2={\left|A_{11}^o\right|}^2={\left|A_{11}\right|}^2& =& {\sin }^2\left(\frac{C}{\sqrt{2}}z\right)& \text{(37)}\end{array}$$

For complete transfer of energy from the ${\mathrm{LP}}_{01}$ mode to the ${\mathrm{LP}}_{11}$ mode, the interaction length, i.e. the length of the twisted core, has to be an odd multiple of the coupling length

$$L_c=\frac{\pi }{\sqrt{2}}\frac{1}{C}.$$

(38)

We derive, now, the modes coupled out from the chiral fiber. From (9) we have

$$\begin{array}{rcll}{e}_{11}^{or,l}(r,\phi )& =& F_{11}(r)\cos \phi (\widehat{x}\mp j\widehat{y})& \text{(39)}\\ e_{11}^{er,l}(r,\phi )& =& F_{11}(r)\sin \phi (\widehat{x}\mp j\widehat{y})& \text{(40)}\end{array}$$

where the $-$ and $+$ in Eq. (39) and (40) correspond to right- and left-polarization, respectively. Combining Eq.(35),( 39 ) and (40), we obtain that the following different forms of ${\mathrm{LP}}_{11}$ modes are coupled out from right - or left-handed structures when right- or left-circularly polarized light beam illuminates the chiral fiber

$$\mathrm{for}\mathrm{right}-\mathrm{handed}\mathrm{helix}:\{\begin{array}{c}\hfill A_{11}(z)F_{11}(r)(\cos \phi -j\sin \phi )(\widehat{x}-j\widehat{y})\mathrm{right}-\mathrm{circularly}\mathrm{polarized}\mathrm{light}\hfill \\ \hfill A_{11}(z)F_{11}(r)(\cos \phi -j\sin \phi )(\widehat{x}+j\widehat{y})\mathrm{left}-\mathrm{circularly}\mathrm{polarized}\mathrm{light}\hfill \end{array}$$

(41)

$$\mathrm{for}\mathrm{left}-\mathrm{handed}\mathrm{helix}:\{\begin{array}{c}\hfill A_{11}(z)F_{11}(r)(\cos \phi +j\sin \phi )(\widehat{x}-j\widehat{y})\mathrm{right}-\mathrm{circularly}\mathrm{polarized}\mathrm{light}\hfill \\ \hfill A_{11}(z)F_{11}(r)(\cos \phi +j\sin \phi )(\widehat{x}+j\widehat{y})\mathrm{left}-\mathrm{circularly}\mathrm{polarized}\mathrm{light}\hfill \end{array}$$

(42)

To analyze the type of modes coupled out from the chiral fiber, it is more convenient to introduce a so-called vortex basis set

$$\begin{array}{rcll}{V}_{11}^{+}(r,\phi )\stackrel{\mathrm{def}}{=}({\mathrm{HE}}_{11}^x+j{\mathrm{HE}}_{11}^y)∕\sqrt{2}& =& (\widehat{x}+j\widehat{y})F_{01}∕\sqrt{2}& \text{(43)}\\ V_{11}^{-}(r,\phi )\stackrel{\mathrm{def}}{=}({\mathrm{HE}}_{11}^x-j{\mathrm{HE}}_{11}^y)∕\sqrt{2}& =& (\widehat{x}-j\widehat{y})F_{01}∕\sqrt{2}& \text{(44)}\\ V_{21}^{+}(r,\phi )\stackrel{\mathrm{def}}{=}({\mathrm{HE}}_{21}^e+j{\mathrm{HE}}_{21}^o)∕\sqrt{2}& =& \exp (j\phi )(\widehat{x}+j\widehat{y})F_{11}∕\sqrt{2}& \text{(45)}\\ V_{21}^{-}(r,\phi )\stackrel{\mathrm{def}}{=}({\mathrm{HE}}_{21}^e-j{\mathrm{HE}}_{21}^o)∕\sqrt{2}& =& \exp (-j\phi )(\widehat{x}-j\widehat{y})F_{11}∕\sqrt{2}& \text{(46)}\\ V_T^{+}(r,\phi )\stackrel{\mathrm{def}}{=}({\mathrm{TM}}_{01}-j{\mathrm{TE}}_{01})∕\sqrt{2}& =& \exp (-j\phi )(\widehat{x}+j\widehat{y})F_{11}∕\sqrt{2}& \text{(47)}\\ V_T^{-}(r,\phi )\stackrel{\mathrm{def}}{=}({\mathrm{TM}}_{01}+j{\mathrm{TE}}_{01})∕\sqrt{2}& =& \exp (j\phi )(\widehat{x}-j\widehat{y})F_{11}∕\sqrt{2}& \text{(48)}\end{array}$$

where we have used the fact that the vector modes may be accurately expressed as linear combinations of $\mathrm{LP}$ modes through the weak guidance approximation. Neglecting the proportional factors, the results can be summarized in the table below

Conclusions

Therefore a single device, i.e. a chiral fiber with a pitch chosen in order to satisfy the phase matching condition and a length of the twisted core designed to be an odd multiple of the coupling length (38), illuminated to one or the other end (left or right handed helix) by circularly polarized light (left or right), is able to produce in output all the elements of the vortex basis. Indeed we obtain pairs of first higher order modes, with a $\pi ∕2$ phase difference between them, that carry both spin angular momentum (SAM) and orbital angular momentum (OAM). The SAM arises due to the orthogonality of the electric fields of these paired modes, while the OAM arises from the azimuthal phase dependence of the beam. It can be shown that a combination of ${\mathrm{HE}}_{21}^e$ and ${\mathrm{HE}}_{21}^o$ modes with $\pm \pi ∕2$ phase shift between them carries one $\hslash$ of SAM and one $\hslash$ of OAM per photon so that the total angular momentum per photon in these modes is $2\hslash$ . A pair of ${\mathrm{TE}}_{01}$ and ${\mathrm{TM}}_{01}$ modes with $\pm \pi ∕2$ phase shift between them carries the same magnitude of SAM and OAM as the ${\mathrm{HE}}_{21}$, but with the SAM and OAM having opposing signs, making the total angular momentum equal to zero.

Furthermore we observe that when a circularly polarized light beam with polarization opposite to the helix handedness illuminates the input end of the chiral fiber, the fundamental ${\mathrm{HE}}_{11}$ fiber mode is converted into an in-phase combination of ${\mathrm{TE}}_{01}$ and ${\mathrm{TM}}_{01}$ modes ( $V_T^{+}$ or $V_T^{-})$, i.e. a set of a azimuthal and radial cylindrical vector beams.