1. Introduction

Because of the possibility of a multifold increase in data-carrying capacity on an optical fiber, research in orbital angular momentum propagation (OAM) mode propagation in a fiber has proliferated. The orthogonality of the OAM modes makes possible the augmentation of fiber capacity by stacking different streams of data on different OAM modes, characterized by a unique set of topological charge, radial index number, and polarization, but possessing the same wavelength. The topological charge, usually denoted $l$, is representative of the orbital angular momentum $l\hbar$ per photon carried by the mode.

The vector OAM modes are the solutions, $HE_{l+1,m}$ and $EH_{l-1,m}$, of the vector wave equation when solved directly in the $e^{\pm il\theta }$ basis [1], where $\theta$ is the azimuthal angle in the cylindrical polar coordinates (this is in sharp contrast to the $\cos l\theta ,\sin l\theta$ basis, traditionally used within the optics community, which by and large treats wave propagation in terms of even and odd modes as in [2]; in this basis, the OAM solutions are subsequently constructed through linear combinations of the even $\cos (l\theta )$ and odd $\sin (l\theta )$ solutions; see, e.g., [3–5]). In what follows, we suppress the radial index $m$ for convenience; however, all numerical calculations assume $m=1$. The transverse fields of the vector OAM modes, $HE_{l+1}$ and $EH_{l-1}$, denoted $\vec {e}^{(l+1)}_t$ and $\vec {e}^{(l-1)}_t$, respectively, satisfy the wave equation

In this paper, we derive the exact analytic expressions for the vector fields in multilayered fibers such as the step-index fiber and the ring-core fiber (Fig. 1), and subject them to close scrutiny for their OAM contents. The structure of the polarization-induced OAM field is especially studied. Subsequently, we ascertain its impact on the polarization correction to the propagation constant of the scalar modes. We begin in Section 2 with the basic definitions and concepts, constructing in the process the general OAM composition of the vector fields of a circular fiber, along with the related concept of the total angular momentum, $J=L+S$. Subsequently (in Section 3), we derive an analytic form of the vector field solutions for the step-index fiber, which is then examined for its OAM contents and features; the extracted features are then numerically simulated and validated. This analysis is followed by a discussion of the extension to the multilayered fiber. The ramifications of the findings are explored in Section 4 for polarization corrections, leading to a general expression for a multilayered fiber (step-index or the three-layered ring-core fiber). This general expression includes the explicit impact of the polarization-induced field component. Analytic expressions specific to the step-index fiber are subsequently derived for illustrative purposes and numerically simulated. Section 5 is the summary and discussion.

 

Fig. 1. a) Cross section of a step-index fiber with core radius $a$ and cladding radius $b~ (>>a); ~n_1$ and $n_2$ are the refractive indices of the core and cladding, respectively; $r$ and $\theta$ along with the $z$ coordinate ($z$ axis coincident with the fiber axis) constitute the polar coordinates; the mode is assumed propagating in the $+z$ direction (out of the plane of the paper). b) Cross section of a ring core fiber, with refractive index $n_1$ for the ring core ($a\le r\le b$) and cladding refractive index equal to $n_2<n_1$ in the fiber region: $0\le r\le a$ and $b\le r\le c~(c>>b)$; in the limit $a\rightarrow 0$, we obtain the step-index fiber.

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2. General OAM composition of the vector fields in a fiber

We begin by writing the generic form of the vector fields expressed directly in the $e^{\pm il\theta }$ basis used in the solution of the vector wave equation:

Similar results are obtained for the associated magnetic field, denoted $\vec {h}_t^{(l\pm 1)}$:

These solutions in conjunction with the $z$ field components, $\vec {e}_z^{(l\pm 1)}$ and $\vec {h}_z^{(l\pm 1)}$, are identified with the familiar vector modes, $HE_{l+1}$ and $EH_{l-1}$, expressed in the exponential basis [14].

In what follows, we determine the exact forms of the two components in Eq. (7) for the step-index fiber and the ring-core fiber (see Fig. 1) using the analytic expressions for the radial and the azimuthal components of the electric fields. In particular, we study the OAM characteristics of the fields in a step-index fiber in detail.

3. OAM modal fields in a multilayered fiber

We consider the step-index fiber (two layers) first followed by an extension of the technique to more than two-layers.

3.1 Step-index fiber

In a standard manner of solving for the fields within a step-index fiber [15], we consider the steady-state solutions: $\vec {E}^{(l)}(\vec {r}, t)=\vec {e}^{(l)}(r,\theta )e^{i(\beta _l z-\omega t)}$ and $\vec {H}^{(l)}(\vec {r}, t)=\vec {h}^{(l)}(r,\theta )e^{i(\beta _l z-\omega t)}$, where $\beta _l$ is the propagation constant of the mode and $\omega =k/\sqrt {\epsilon _0 \mu _0}$ is the angular frequency. We subsequently apply the Maxwell’s equations, starting with $E_z^{(l)}(r,\theta ) = e_z^{(l)}(r)e^{il\theta }$, where $e_z^{(l)}(r)=A_lJ_l(p_lr)$, and $H_z^{(l)}(r,\theta ) = h_z^{(l)}(r)e^{il\theta }$, where $h_z^{(l)}(r)=B_lJ_l(p_lr)$, within the fiber core of radius $a$; similarly, for the cladding region ($r\ge a)$, we write $E_z^{(l)}(r,\theta ) = e_z^{(l)}(r)e^{il\theta }$, where $e_z^{(l)}(r)=C_lK_l(q_lr)$, and $H_z^{(l)}(r,\theta ) = h_z^{(l)}(r)e^{il\theta }$, where $h_z^{(l)}(r)=D_lK_l(q_lr)$. The results for the radial profiles, $e_r^{(l)}$ and $e_\theta ^{(l)}$, comprising the transverse field (see Eq. (5)) are

3.1.1 Analysis of the $J=l+1$ case

Changing $l$ to $l+1$ in Eq. (13) and inserting the resulting expressions in Eq. (7), we obtain for the fields within the core

Equation (15) is a very general result, and a remarkable one because the spatial forms in Eq. (15) resemble the scalar field amplitudes (solutions of the scalar wave equation). Parameter $p_{l+1}$ in the argument of the Bessel functions, $J_l$ and $J_{l+2}$, is given by $p^2_{l+1}=k^2n_1^2-\beta _{l+1}^2$. The corresponding counterpart $\tilde {p}_{ l}$ in the solution of the scalar wave equation, $\vec {\tilde e}^{(l)}_t\sim J_{l}(\tilde {p}_l r)e^{ il\theta }\vec {\epsilon }_{+}$ (see Eq. (2)), is similarly related by $\tilde {p}_l^2=k^2n_1^2-\tilde {\beta }_l^2$. Consequently, $\tilde {p}_l^2- p_{l+1}^2=\beta _{l\pm 1}^2 - \tilde \beta _{l}^2= \delta \beta _{l\pm 1}^2$, which is of order $\Delta <<1$ in WGA and therefore very small. Thus, $p_{l+1}$ in the argument of $J_l$ in Eq. (15) can be replaced with $\tilde {p}_{ l}$, implying we may identify the first term in Eq. (15) with the field of the traditional scalar $OAM_l$ mode possessing a topological charge $l$ and an amplitude $O_{l}(r,\theta ) =J_{l}(\tilde {p}_l r)e^{ il\theta }$ [1] (see also Appendix A). Peak intensity defined as the square of the magnitude of $O_{l}$ occurs at the first maximum of the $J_l (\tilde {p}_l r)$ function. If we define $r=r^{(l)}_{max}$ as this maximum, then $r^{(l)}_{max}=x_{max}/\tilde {p}_l$, where $x_{max}$ is the first maximum of $J_l(x), l>0$. Since the Bessel function rises from its zero value with increase in its argument value, the intensity of this mode as a function of the radial distance $r$ is the familiar donut pattern, often observed in experimental data (see, e.g, [5]).

The second term in Eq. (15), however, does not correspond to the field of the traditional scalar $OAM_{l+2}$ mode because its radial profile, $J_{l+2}(p_{l+1}r)\approx J_{l+2}(\tilde {p}_{l}r)$ is not the same as the profile, $J_{l+2}(\tilde {p}_{l+2}r)$ of the traditional $OAM_{l+2}$ scalar mode. This is due to the fact that $\tilde {p}_{l+2}\ne \tilde {p}_{l}$. In fact, $\tilde {p}_{l+2}> \tilde {p}_{l}$ (see Table 1) because $\tilde {\beta }_{l+2}< \tilde {\beta }_{l}$. As a result, the second term in Eq. (15) is a modified $OAM_{l+2}$ modal field, whose predicted intensity maximum occurs at $r=r'^{(l+2)}_{max}$, where

 

Fig. 2. The $J=l+1$ case; the intensity pattern within the core, $J^2_{l+2}(\tilde {p}_{l+2}r)$, corresponding to the traditional scalar $OAM_{l+2}$ mode (on the left) and the expanded out intensity pattern ($J_{l+2}^2(\tilde {p}_lr)$) corresponding to the modified scalar $OAM_{l+2}$ modal field (on the right), as predicted by our theoretical expression (the second term in Eq. (15)); see text for more details; the top row is for $l=3$ and the bottom row is for $l=5$ (see Table 1).

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Table 1. The second term in Eq. (15) is the field of a nontraditional (modified) scalar $OAM_{l+2}$ mode, whose intensity maximum at the radial distance, $r'^{(l+2)}_{max}$ is shifted outward by a factor $f^+_l= \tilde {p}_{l+2}/\tilde {p}_{l}$ relative to the position $r^{(l+2)}_{max}$ of the intensity maximum of the conventional scalar $OAM_{l+2}$ mode; the numerical values are for a step-index fiber with parameters: $n_1=1.461,~n_2=1.444,~ a=25 \mu m,\lambda =1.55 \mu m$ (see Fig. 1).

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3.1.2 Analysis of the $J=l-1$ case

Changing $l$ to $l-1$ in Eq. (13) and substituting in Eq. (7), we obtain

The field corresponds to the $EH_{l-1}$ mode. In WGA, $\beta _{l-1}\approx \tilde {\beta }_l$, the corresponding solution of the scalar wave equation (see Section 1); consequently, $p_{l-1}\approx \tilde {p}_l$. So, the first term corresponds to the traditional scalar mode with the radial profile, $J_{l}(\tilde {p}_{l}r)$, while the second term is a modified scalar modal field because the associated radial function, $J_{l-2}(p_{l-1}r)\approx J_{l-2}(\tilde {p}_l r)$, is different from $J_{l-2}(\tilde {p}_{l-2})r$, the radial profile of the scalar OAM mode characterized by topological charge $l-2$ . The donut intensity pattern here has a radius

 

Fig. 3. The $J=l-1$ case; the intensity pattern within the core, $J_{l-2}^2(\tilde {p}_lr)$, corresponding to the traditional scalar $OAM_{l-2}$ mode (on the left) and the shrunken intensity pattern $J^2_{l-2}(\tilde {p}_{l-2}r)$ corresponding to the modified scalar $OAM_{l-2}$ field (on the right), as predicted by our theoretical expression (the second term in Eq. (18)); see text for more details; the top row is for $l=3$ and the bottom row is for $l=5$ (see Table 2).

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Table 2. The second component in Eq. (18) is a nontraditional (modified) scalar $OAM_{l-2}$ modal field, whose intensity maximum, $r'^{(l-2)}_{max}$ is shifted by a factor $f^-_l= \tilde {p}_{l-2}/\tilde {p}_{l};~r^{(l-2)}_{max}$ is the position of the intensity maximum of the conventional scalar $OAM_{l-2}$ mode; the numerical values are for a step-index fiber with parameters: $n_1=1.461,~n_2=1.444,~ a=25 \mu m,\lambda =1.55 \mu m$ (see Fig. 1).

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3.1.3 Relative amplitude of the modified scalar modal field

We need to evaluate the coefficients, $\gamma ^{(l+1)}_\pm$ (see Eqs. (15) and (16)) and $\gamma ^{(l-1)}_\pm$ (see Eqs. (18) and (19)). We first note that

In the limit, $n_2\rightarrow n_1$, the above characteristic equation reduces to the characteristic equation for the scalar modes [2,15], assuming $\beta _l\approx kn_1$. In a real fiber, $n_2$ and $n_1$ are not exactly equal, being related by $n_2^2=n_1^2(1-2\Delta )$, where $\Delta$ is usually a small fraction. Inserting this expression in Eq. (22) then leads to the following relationship:

In terms of the above parameters, Eq. (21) can be rewritten as

We change $l$ to $l+1$ in Eq. (28) and insert the resulting expression for $B_{l+1}/A_{l+1}$ with a negative sign, in Eq. (16) to obtain

When $\Delta ~<<1$ (e.g., $\Delta \approx 0.01$ in a standard commercial multimode step-index fiber and $\approx 0.02$ in considered ring-core fibers [6,7]), we can expand Eqs. (29) and (30) to first order in $\Delta$; further, using the fact that $\beta _{l+1}\approx kn_1$, we arrive at

Here we change $l$ to $l-1$ in Eq. (28) and insert the resulting expression (with the positive sign) in Eq. (19). This yields the following expressions:

Retaining the first term in the binomial expansion of Eqs. (34) and (35) in $\Delta (<<1)$, and using the fact that $\beta _{l-1}\approx kn_1$, we arrive at

3.2 Numerical validation

Here we compare the exact solution of the vector wave equation with the results derived on the basis of $\Delta <<1$; in the latter case, we recast the results in terms of the solutions of the scalar wave equation. Consequently, focusing first on the $J=l+1$ case, we insert Eqs. (31) and (32) (or (33)) into Eq. (15), and set $\beta _{l+1}\approx \tilde {\beta }_l$, $p_{l+1} \approx \tilde {p}_l$, and $q_{l+1}\approx \tilde {q}_l$, results valid in WGA. Subsequently, we arrive at

For our calculations done in MATLAB in double precision, we consider a multimode fiber with the same parameters as those used in Tables 1 and 2. Table 3 shows the results of numerical integration done for $I^{'(J)}_V$ and $I^{'(J)}_S$, where $J=l\pm 1$; for the former, we used two different characteristic equations, one corresponding to the $HE_{l+1}$ modes and the other to the $EH_{l-1}$ modes, while for $I^{'(J)}_S$, we used the single characteristic equation corresponding to the scalar modes [2]. The differences are of the order of a tenth percent or less, corroborating the veracity of our results expressed in terms of the scalar parameters. Furthermore, we observed that, using parameter $\tilde {p}_{l+2}$, instead of $\tilde {p}_l$, in the argument of $J_{l+2}$ in Eq. (39), causes %diff in Table 3 (for the $l+1$ case) to increase by a factor of 2 to 9 in magnitude, supporting the fact that the polarization-induced component of the electric field is indeed the field of a modified scalar OAM mode of topological charge $l+2$ as described in Section 3.1, and not the field of a traditional scalar mode of the same topological charge. Similar worsening of the results (although less dramatic) occurs in the $J= l-1$ case, when $\tilde {p}_l$ in the argument of the Bessel function, $J_{l-2}$, is replaced with $\tilde {p}_{l-2}$.

Table 3. $I^{'(J)}_V=I_V/(2\pi |A_{J}|^2)$ and $I^{'(J)}_S=I_S/(2\pi |A_{J}|^2)$ for the $J=l\pm 1$ cases and various values of topological charge $l$; the percentage difference (%$diff$), defined as $(I^{'(J)}_V-I^{'(J)}_S)/I^{'(J)}_V x 100$, is of the order of 0.1 or less in magnitude (see text for more details); $n_1=1.461, n_2=1.444, a =25 \mu m, \lambda =1.55 \mu m$.

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3.3 Extension to a multilayered fiber

In extending the above theoretical concepts to a cylindrical waveguide with more than two layers, we observe that, within each layer of the waveguide, the field solutions are, in general, a linear combination of the Bessel functions: $J_l(\alpha _i r)$ and $Y_l(\alpha _i r)$, when $\alpha _i^2=(k^2n_i^2-\beta _l^2)$ is positive, and a linear combination of the modified Bessel functions: $I_l(\alpha _i r)$ and $K_l(\alpha _i r)$, when $\alpha _i^2$ is negative. For the 3-layer fiber, defined by a ring core of radii $a$ and $b (>a)$, refractive index $n_1$, and inner and outer cladding refractive index of $n_2<n_1$ (Fig. 1), the z component of the field is expressible in the azimuthal exponential basis as

Within the ring, $a\le r\le b$, where the fields are primarily concentrated, we obtain, using Maxwell’s equations (as in the step-index fiber case) and the appropriate Bessel function recursion relations, the following expression:

4. General form of the polarization correction

In this section, we evaluate the impact of the second component (the modified scalar modal field) on the polarization correction to the propagation constant of a scalar mode. Prior work [2,8–10] has not considered it, partly due to the lack of an analytic expression as discussed in Section 1. We first derive a generic form for polarization correction of propagation constant in a mutlilayered fiber. This form includes the explicit contribution of the polarization-induced component (the second component) in Eq. (7). We also assume the WGA so that we can write the polarization-induced field correction (see Eq. (4) and Eq. (7)) as

For multilayered fibers such as the step-index fiber and the three-layer ring-core fiber, characterized by a core refractive index of $n_1$ and a cladding refractive index of $n_2$ (see Fig. 1), the function $G_{l\pm 2}(r)=G_{l\pm 2}(p_{l+1}r)$, where the parameter $p_{l+1}^2=k^2n_1^2-\beta _{l+1}^2$. Recalling $\vec {\tilde {e}}_t^{(l)}=F_{l}(r)e^{il\theta }\vec {\epsilon }_\pm$, we substitute $\vec {e}^{(l\pm 1)}_t= \vec {\tilde e}^{(l)}_t + \delta \vec {e}^{(l\pm 1)}_t$ into Eq. (1). After taking scalar products and performing the necessary integrations (see Appendix B), we arrive at the general form for polarization correction

The radial amplitudes, $F_l(r)$ and $G_{l\pm 2}(r)$, are characteristic of the multilayered fiber under consideration. The first integral is identified with the familiar polarization correction [2,8,10]; the second integral is the correction due to the polarization-induced field component. Furthermore, $\Delta$ being less than one allows us to make an expansion of $ln(n^2)$ in the parameter $\Delta$ (Section 1). The function $G_{l\pm 2}$ in the second term of Eq. (45) is of order $\Delta$, so an expansion of $ln (n^2)$ to first order in $\Delta$ yields a result quadratic in $\Delta$.

4.1 Expression valid up to second order in $\Delta$

To obtain the expression valid up to second order in $\Delta <<1$, we note that $n^2=n_1^2(1-2\Delta f(r))$ and use the expansion, $ln(1-x)=-x-x^2/2+\cdots$ in Eq. (45) to arrive at

If we ignore the second term and replace $1+2\Delta f(r)$ by unity in Eq. (46), we obtain the result in [2,10], which is the correction to first order in $\Delta$. We consider now, for illustrative purposes, the step-index fiber.

4.2 Application to a step-index fiber

$n^2=n_1^2(1-2\Delta f(r))$, where $\Delta <<1$. As a consequence, to first order in $\Delta$, $\partial ln(n^2)/\partial r=-2\Delta \partial f(r)/\partial r=-2\Delta \delta (r-a)$, the last step following from the fact that $f(r)$ is a step function at $r=a$. Therefore, Eq. (46) reduces to

4.3 Numerical results

For numerical simulation we chose the multimode step-index fiber with parameters: $n_1=1.461, n_2=1.444, a=25 \mu m$, and $\lambda = 1.55 \mu m$. The index profile height parameter $\Delta =0.011568$. We first calculated $\delta \beta _{l+1}^{(CE)}=\beta _{l+1}-\tilde {\beta _l}$ and $\delta \beta _{l-1}^{(CE)}=\beta _{l-1}-\tilde {\beta }_l$, where $\beta _{l+1}$ and $\beta _{l-1}$ were determined from the characteristic equations for the $HE_{l+1}$ and the $EH_{l-1}$ modes, respectively, while the propagation constant, $\tilde {\beta }_l$ was determined from the characteristic equation for the scalar modes [2]. Values are displayed in Table 4 for $l=2$ to $l=5$. $\delta \beta _{l\pm 1}$, calculated directly from Eq. (50), is in agreement with the characteristic equation-based values $\delta \beta _{l\pm 1}^{(CE)}$ within a fraction of a percent. The fifth and the ninth columns are the contributions of the polarization-induced OAM components, denoted $\delta \beta _{l\pm 1}^{(p)}$, which is calculated from the second term in Eq. (50). It is observed to be about 1% of the total, which is of the order of $\Delta$, as expected. The fractional differences (%$diff$) are likely due to higher order terms not considered here. All numerical calculations were done in MATLAB in double precision.

Table 4. Polarization corrections, $\delta \beta _{l\pm 1}^{(CE)}$ (in units of $meter^{-1}$) as calculated from the characteristic equations [2], and $\delta \beta _{l\pm 1}$ as calculated directly from the analytic expression, Eq. (50); the differences are of the order of a tenth of a percent; the fifth and the ninth columns are the contributions of the second term in Eq. (50).

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4.4 Extension to ring-core fiber

A similar calculation can also be carried out for the ring-core-fiber (see Fig. 1) once the function $G_{l\pm 2}(r)$ has been extracted from the expression, Eq. (43); the refractive index function here is given by $n^2(r)=n_1^2(1-2\Delta f(r))$, where $f(r)=g(r-a)+h(r-b); g(r-a)= 1$ for $r<a$ and equal to zero for $r\ge a$, and $h(r-b)=1$ for $r\ge b$ and equal to zero for $r<b$. For $\partial f(r)\partial r$ needed in Eq. (46), we note here that $\partial h(r-b)/\partial r= \delta (r-b)$ and $\partial g(r-a)/\partial r =-\delta (r-a)$.

5. Summary and discussion

We have critically investigated the nature of the transverse OAM fields in a multilayered cylindrical waveguide. In particular, in a detailed study of the step-index fiber (two-layered fiber), we have identified the polarization-induced component of the vector modal field as a modified scalar OAM mode, a hitherto unknown result. The intensity peak of this nontraditional OAM mode occurs at a larger or smaller radial distance in the characteristic donut intensity pattern, depending upon whether the spin is aligned with OAM or antialigned with OAM as in the transverse fields of the $HE_{l+1}$ or the $EH_{l-1}$ mode, respectively. An extension to other multilayered fibers such as the ring-core fiber shows the modified scalar field as a general feature of the polarization-induced component. The amplitude of this modified scalar mode is of order $\Delta$, the index profile height parameter; thus larger the value of $\Delta$, larger this amplitude. We also have developed analytic expressions for the contribution of this component to polarization correction, not done before. Specifically, we have applied these expressions to the case of a step-index fiber to illustrate their impact numerically. Such expressions along with the insight they provide can be useful in the analysis and design of fibers. Note also that, while the treatment here pertains to the multilayered fibers, the result of a polarization-induced component being a modified scalar modal field is a generic result that will manifest in any type of fiber (graded-index, or otherwise), simply because $\tilde {p}_l \ne \tilde {p}_{l\pm 2}$ in general.