## Abstract

A new type of interaction between optical waves occurs in chirally-coupled-core (CCC) fibers. Instead of linear-translational symmetry of conventional cylindrical fibers, CCC fibers are helical-translation symmetric, and, consequently, interaction between CCC fiber modes involves both spin and orbital angular momentum of the waves. Experimentally this has been verified by observing a multitude of new phase-matching resonances in the transmitted super-continuum spectrum, and theoretically explained through modal theory developed in helical reference frame. This enables new degrees of freedom in controlling fiber modal properties.

© 2011 OSA

## 1. Introduction

Optical fibers are critically important in a wide range of applications such as telecommunications [1], biomedical imaging and surgical procedures, frequency-standard metrology [2], optical sensors, and currently are enabling revolutionary advances in new generation of scientific and industrial lasers. The key advantage of optical fibers as well as the main reason for being so broadly used is due to the paraxial wave guidance, which produces practically lossless light propagation for long distances while maintaining a fixed and well-defined wavefront. There are two methods of achieving paraxial wave guidance based on either total internal reflection [3] or photonic-crystal bandgap [4], and, up to now, fiber modal properties were controlled only by tailoring the confinement of such guided waves. Here we introduce Chirally-Coupled-Core (CCC) fiber structures, in which guided modes can interact with each other involving their optical angular momentum [5], thus, as we discovered, providing with fundamentally new degrees of freedom in controlling modal properties of optical fibers.

Optical angular momentum plays an important role in defining properties of this composite CCC waveguide, because their modes are invariant with respect to helical-translation (instead of independent rotational and linear-translation invariance of conventional fibers), and, therefore, inherently possess both orbital- and spin- optical angular momentum. One interesting example of novel modal properties is the possibility of designing CCC fibers which support a single transverse mode in an arbitrarily large diameter strongly-guiding core, thus overcoming one of the fundamental limitations of conventional fibers. This can be achieved because the fundamental mode has only spin-angular momentum while all LP_{nm} higher-order modes (except LP_{0m} modes) have orbital-angular momentum as well. This leads to symmetry-based distinction between modes allowing the implementation of structures where all modes except the fundamental one are radiation modes and, therefore, do not propagate. Another example is the possibility of tailoring spectral transmission properties to control nonlinear interactions of high-intensity beams propagating in such fibers [6]. Furthermore, beams with well defined optical spin-angular momentum (circular polarization) [5,7], and, more recently, with well defined optical orbital angular momentum (optical vortex) [5,8] have found a number of interesting applications including particle trapping and manipulation [9,10], quantum communication [11], and quantum computing and information encoding in multi-dimensional quantum space [12]. Therefore, the possibility of controlling orbital-angular (as well as spin-angular) momentum of CCC fiber modes is also interesting.

## 2. CCC fiber geometry and quasi-phase-matched interactions between modes in the structure

CCC fiber structure shown in Fig. 1a
consists of two wave-guiding cores deposited within one glass cladding and placed in optical proximity, so that they form a weakly coupled waveguide system. Fabrication of such a structure is relatively straightforward – it requires a fiber perform with both cores running straight through it, central core being on axis and side-core off-axis. Spinning such a fiber preform during fiber draw process using conventional fiber-preform spinning techniques will then produce this CCC structure. The core in the center of the cladding referred to as “central core” runs straight along the fiber axis, as in ordinary fibers. It is this core that serves for transmitting or amplifying the signal, just as in conventional fibers. The other core is referred to as “side core” and is chirally winding around the center core following a helical path with a constant “helix pitch *Λ*” and “offset *R*” (Fig. 1b). It is typically much smaller in diameter and its primary purpose is to control modal properties of the center core.

Indeed, it is well known that, in a conventional weakly-coupled parallel waveguide system, modes interact only when their phase velocities are matched (*i.e.* linear momentum or, equivalently, modal propagation constants are equal [13]). If the two parallel waveguides are different (different refractive indices or sizes), then for each interacting modal pair such phase-matching could be achieved only at a single wavelength, when the dispersion curves of the two waveguides cross [13]. However, in CCC fibers with a helical side core, groups of multiple and periodically-spaced mode-coupling resonances are experimentally observed in spectral domain for each interacting modal pair. These resonances are equally separated from each other by the helix rotation rate *𝐾* = 2*𝜋/Λ* in terms of propagation constants of the modes, and only one of the resonances corresponds to the exact phase-matching condition. In terms of a physically intuitive picture, these multiple resonances can be explained by a relative phase accumulation between different modes due to their angular momentum difference. Relative phase difference between the modes is accumulated due to the relative “slippage” of the phases of rotating fields with its sign depending on whether this rotation has the same or opposite sense of rotation with respect to the chirality of the CCC helix. Because in this structure modal interactions occur when phase-velocity mismatch is compensated by angular momentum difference, such interactions should be considered to be quasi-phase-matching (QPM).

Figure 2
presents the detailed description of this experimental observation for the particular CCC fiber sample described in Fig. 1a. In Fig. 2a, wavelength-dependent loss for side-core LP_{11} and LP_{21} modes due to the frustration of the total internal reflection in a curved waveguide [14] is calculated [15]. Figure 2b shows the curves of calculated modal refractive indices for side-core modes LP_{11} and LP_{21}, and central-core fundamental mode LP_{01}, together with which two groups of vertical dashed color lines are plotted to represent QPM resonance positions: the blue group on the left for LP_{21} and the red group on the right for LP_{11}. These resonances are calculated from Eq. (13) and labeled with parameter$\Delta m$, which are derived from our theoretical model (see details below). In Fig. 2c, experimentally measured transmission spectrum has been obtained by launching a broad-band super-continuum laser source into the central 35µm-diameter core of the 1.5m-long CCC fiber. Main observation here is that multiple resonances appear as transmission “dips”. Most important, the extension of these “dips” marked as the vertical light-grey dotted lines appear to exactly overlap with the calculated QPM resonances in Fig. 2b. This verifies the fact that QPM coupling involving optical angular momentum indeed occurs. Further comparison between Fig. 2a, 2b and Fig. 2c reveals that modal-interaction resonances are observed as transmission “dips” only at the wavelengths where a side-core mode experiences non-negligible losses, namely exceeding approximately 1dB/m loss. Modal interactions at wavelengths shorter than these 1dB/m-loss wavelengths (marked in Fig. 2b by points *A* and *B* for side-core LP_{21} and LP_{11} modes respectively) still do occur, but do not result in observable central-core modal loss. This has been experimentally verified for the “lossless” resonance at 1030nm (see discussion of Fig. 3
below). In addition, Fig. 2d shows numerically calculated transmission spectra of all central-core modes, and the comparison between Fig. 2c and Fig. 2d shows that all these “dips” belong to central-core fundamental mode, which also indicates the effective single-mode operation of this particular CCC sample (see details below).

## 3. Maxwell’s equations in curvilinear helical reference frame

Challenge of describing CCC structure modes and their interactions is associated with the fact that the cross-sectional profile of the structure rotates with the translation along the fiber axis. In particular, this complicates description of side-core modes in a Cartesian reference frame {*x*, *y*, *z*}, whose z-axis is oriented along the central axis of the fiber, since position and orientation of the side core is z dependent in this reference frame. However, mathematical description of QPM modal interactions in a CCC structure becomes simple and elegant when derived in a curvilinear helical reference frame, whose x- and y-axes are rotating around the fiber axis with the periodicity *Λ* of the side-core helix [16]. This reference frame captures inherent helical symmetry of a CCC structure and its interacting modes, since in these helical coordinates CCC fiber is represented by a longitudinally-invariant refractive index with both cores straight and parallel to each other, *i.e.* the helical winding of a side core becomes “unwound” [16,17]. Furthermore, in this reference frame Maxwell’s equations can be expressed in a mathematical form identical to that in Cartesian coordinates and the only difference between the two descriptions is solely captured by the transformed dielectric permittivity and magnetic permeability tensors expressed in this helical frame [16,17]. For example, a conventional cylindrically-symmetric step-index fiber, which is piece-mall homogeneous and isotropic in Cartesian coordinates, in helical coordinates is described by inhomogeneous and anisotropic permittivity and permeability tensors [17]. Following this approach, we can obtain the eigenmodes of this CCC structure, and then describe QPM interactions between these modes.

Helical coordinates {*X*, *Y*, *Z*} are related to Cartesian coordinates {x,y,z} through:

*K*is the helical reference frame’s rotation rate.

For simplicity let’s assume that CCC fiber cladding is homogeneous and isotropic, and both the central core and side core are isotropic step-index profiles. Then, in the Cartesian reference frame {*x*, *y*, *z*}, the dielectric permittivity tensor $\widehat{\epsilon}\left(x,y,z\right)$ of the CCC structure is expressed as:

In general, it can be shown [16,17] that from the covariance of Maxwell’s equations it follows that in any generalized coordinate system these equations can be expressed in a mathematical form identical to their expression in Cartesian coordinates, reference-frame difference being solely captured by the change in the tensor form of permittivity and permeability in these equations:

*X*and

*Y*are rotating together with the side core, the longitudinal variation of the side core disappears in the helical coordinates and we can choose ${\Delta {\epsilon}_{2}(X,Y)=\Delta {\epsilon}_{2}(x,y,z=0)|}_{x\to X,y\to Y}$. Consequently, the component ${\widehat{\epsilon}}_{straight}^{h}\left(X,Y\right)$ describes dielectric-permittivity distribution of straight center and side cores. The effects associated with rotation of the CCC structure now are solely captured by the component ${\widehat{\epsilon}}_{rotate}^{h}\left(X,Y\right)$. Inspection of Eq. (6) reveals that the dielectric permittivity tensor ${\widehat{\epsilon}}^{{}_{h}}\left(X,Y\right)$, consisting of both components, becomes

*z-coordinate invariant*(although it becomes anisotropic due to component ${\widehat{\epsilon}}_{rotate}^{h}$), which verifies the intuitive expectation that in helical coordinates the CCC fiber geometry should be “unwound” and represented by a straight,

*Z*-axis independent (but inhomogeneous and anisotropic) waveguide structure.

This establishes the basic framework for numerical and analytic study of CCC structures. For example, the obtained Z-invariance of transverse permittivity-tensor distribution in Eq. (4) enables to apply standard finite-element-method (FEM) modal solvers for anisotropic waveguides and to numerically find exact modal profiles and complex propagation constants (imaginary parts describing modal loss) of either individual CCC-structure waveguides or composite-modes of the complete structure. Examples of numerically calculated CCC-structure modes are shown in Fig. 1e.

## 4. Helically symmetric eigenmodes in CCC structure

In order to describe QPM conditions in CCC structures it is first necessary to consider CCC-structure modes in this helical reference frame. The main result here is that helical-translation symmetry of a CCC structure constrains allowable interacting individual-waveguide modes of the structure to only those that are helically-invariant. This is equivalent to requiring that modal fields must be invariant *simultaneously* to both translation along Z-axis and rotation around the Z-axis. As it is shown in detail below, this constraint leads to modal-field vector-direction distributions (see Fig. 1c) and vector-amplitude distributions (see Fig. 1d) that make full rotation with each optical cycle, which are associated with optical spin-angular (*i.e.* circular-polarization) momentum and optical orbital-angular momentum respectively. This also produces additional phase shifts per unit propagation length, adding to the phase shift due to field propagation along fiber axis Z.

This can be shown using helical-coordinate approach outlined in the previous section. The examination of the helical-frame dielectric-permittivity tensor ${\widehat{\epsilon}}^{{}_{h}}$ consisting of ${\widehat{\epsilon}}_{straight}^{h}$ and ${\widehat{\epsilon}}_{rotate}^{h}$in Eqs. (5) and (6) reveals that the magnitude of tensor ${\widehat{\epsilon}}_{rotate}^{h}$components at each transverse point (*X*, *Y*) is usually much smaller than the magnitude of tensor ${\widehat{\epsilon}}_{straight}^{h}$ components, because the rotation of CCC structure is relatively “slow” due to helix pitch Λ in CCC structures being typically between 5mm – 10mm, and the helix offset *R* for the side core (associated with the largest values of coordinates *X* and *Y* in Eq. (6)) typically in tens of micrometers. This enables to first solve for the modal fields in straight and isotropic fiber cores by retaining only tensor ${\widehat{\epsilon}}_{straight}^{h}$, which for round step-index cores can be done analytically [18], and then to examine contribution from ${\widehat{\epsilon}}_{rotate}^{h}$ as a weak perturbation, for example, by employing Coupled Mode Equations (CME).

For round, cylindrically symmetric fibers with step index profile, the analytical modal solutions for vector modal fields are well known [19]: consisting of Transverse Electric modes designated as $T{E}_{0m}$, Transverse Magnetic modes designated as $T{M}_{0m}$, and so called Hybrid modes designated as $H{E}_{nm}$ and $E{H}_{nm}$, with indices *n* and *m* standing for azimuth and radial modal numbers *n* = +1, +2, … and *m* = +1, +2, … Each $H{E}_{nm}$ or $E{H}_{nm}$ mode is doubly-degenerate with respect to two orthogonally-oriented electric-field’s azimuthal distributions. The electric field azimuthal distributions for each set of degenerate $H{E}_{nm}$modes are [18–20]:

*EH*mode set are [18–20]:

It can be shown that the perturbation due to the term ${\widehat{\epsilon}}_{rotate}^{h}$ will cause coupling between the degenerate pair of straight fiber modes, such as $H{E}_{nm}^{o,e}$ degenerate modal pair and $E{H}_{nm}^{o,e}$ degenerate modal pair, and this coupling is described by the following CME for the n-*th* order modal pair:

*n*-th order degenerate pair. Subscripts “c” and “s” stand for “central core” and “side core” respectively, since each core has its own set of CME. These coupled-mode equations were rigorously derived directly from Maxwell equations Eq. (3) in helical reference frame using standard methods, for example the method in Ref [18]. From this derivation it follows that there is no contribution from ${\widehat{\epsilon}}_{rotate}^{h}$ to propagation-constant values ${\beta}_{c}^{n}$ for the central core (which remains the same as for the straight core), while the effect of ${\widehat{\epsilon}}_{rotate}^{h}$on the off-center side core is to produce a helical correction factor ${\beta}_{s}^{n}\cong {\beta}_{straight}^{n}\cdot \sqrt{1+{K}^{2}{R}^{2}}$, which accounts for the helical path that side-core takes along the z-axis direction. By solving the eigenvalue problem of the CME in Eq. (9), we obtain helically-symmetric modal solutions for the

*n*-th order modes in both central and side core expressed as the following combination of straight-waveguide modes:

The two modes in this set are designated by the superscript “±”, where “+” and “-“ here refer to the same and the opposite senses of rotation respectively, with respect to the helical reference frame (or, equivalently, CCC structure) chirality. These two modes, formed by mixing two degenerate “static” straight-waveguide modes, are not degenerate in the sense that they each acquire different propagation-constant increments $\Delta \beta =\pm nK$ (for the n-*th* order modes).

By direct calculation it can be shown that electric-field’s azimuthal dependence for $H{E}_{nm}^{\pm}$ modes is:

*X, Y*and

*Z*axes respectively, and $\theta $ is the azimuth angle. It is well known that the term ${e}^{\pm jl\theta}$ describes an optical vortex carrying optical orbital angular momentum (OAM) [5,8,22], and the term ${\widehat{e}}_{\pm}={\widehat{e}}_{x}\pm j{\widehat{e}}_{y}$ describes circularly polarized light carrying spin-angular momentum (SAM) [5,7,22]. Propagation constant increment $\Delta \beta $ due to OAM is $\pm (n-1)K$ for $H{E}_{nm}^{\pm}$ modes and $\pm (n+1)K$ for $E{H}_{nm}^{\pm}$ modes, and similarly additional propagation constant increment $\Delta \beta $due to SAM is $\mp K$ for $H{E}_{nm}^{\pm}$ modes and $\mp K$ for $E{H}_{nm}^{\pm}$ modes. By comparing Eqs. (11, 12) with Eq. (10), one can recognize that the summation of OAM-induced $\Delta \beta $ and SAM-induced $\Delta \beta $ produces the total angular momentum (OAM + SAM) contribution to the modal propagation constant that is always equal to $\pm nK$ for

*n*-th order $H{E}_{nm}^{\pm}$ or $E{H}_{nm}^{\pm}$ modes. Using FEM modal solver in a helical reference frame has confirmed that total angular momentum contribution $\Delta \beta $to the propagation constants of$H{E}_{nm}^{\pm}$ and $E{H}_{nm}^{\pm}$ modes is always equal to $\pm nK$, consistent with the theoretical derivation. These results are summarized in Table 1 . Note that these additional phases are relative phases of each mode with respect to a different mode of a CCC structure, but not with respect to a “laboratory” reference frame,

*i.e.*are only important for describing coupling between different non-degenerate modal pairs.

Note, that n = 0 corresponds to $T{E}_{0m}$ modes and $T{M}_{0m}$ modes which are azimuthally invariant but with no optical angular momentum, and that modes with n = 1 (which include fundamental HE_{11} modes) carry only spin-angular momentum.

## 5. Quasi-phase matching in a CCC structure

For mathematically expressing QPM conditions it is convenient to group $T{E}_{0m}$modes,$T{M}_{0m}$ modes, and$H{E}_{nm}$ modes and $E{H}_{nm}$ modes into so-called $L{P}_{lm}$ modes ($LP$stands for “*Linearly Polarized*”) as shown in Fig. 2. The $L{P}_{lm}$ modes are customarily used in designating weakly-guiding cylindrically-symmetric fiber modes [19,20], which are groups of $T{E}_{0m}$ modes, $T{M}_{0m}$ modes, and $H{E}_{nm}$and $E{H}_{nm}$ hybrid modes that become degenerate (their propagation constants *β* are equal) in the weakly guiding limit [21].

This grouping into $L{P}_{lm}$ modes in a CCC structure remains valid and has been summarized in Table 2 , where helically-symmetric $H{E}_{nm}^{\pm}$ and $E{H}_{nm}^{\pm}$ modes are used. However, it is important to note that there is an essential distinction between the $L{P}_{lm}$ modes in CCC structures and in conventional cylindrical fibers. Indeed, by comparing Table 1 and Table 2, one can see that the degeneracy within the $L{P}_{lm}$ mode group in CCC structures is broken in the helical reference frame, and when they interact with each other, the modes carrying different orbital- and spin-angular momentum combinations (in terms of magnitudes and polarities) in one $L{P}_{lm}$ mode group in CCC structures will act as they are not degenerate any more. Of course, when no interaction between the cores of CCC structure is considered (namely outside quasi-phase matched resonances), modal degeneracy within each core is still valid.

In order to determine QPM conditions between modes in a CCC structure, we have developed a rigorous analytical theory based on Maxwell’s equations in helical reference frame, with which we have formulated the CME describing inter-core coupling between different helically-symmetric vector modes from Table 1 and Table 2, and derived exact QPM conditions for each interacting modal pair. The derived expressions for these helically-symmetric vector modes are somewhat complicated. However, we noticed that the QPM expressions could be quite simple if interacting modes are grouped according to *LP*-grouping prescription in Table 2. Essentially, all possible QPM conditions of coupling between $L{P}_{lm}$ modal groups involve all possible combinations of orbital- and spin- angular momentum values and polarities within each $L{P}_{lm}$ group. In terms of propagation constant and angular momentum matching, this can be expressed as following:

Physical origin of the multitude of QPM resonances described by Eq. (13) is associated with three different types of perturbations that are causing three different types of coupling between center and side core modes. We have revealed this by developing a coupled-mode theory in a helical reference frame as well as performing detailed finite-element analysis of induced birefringence in CCC fiber, and by comparing theoretical results with the experimental observations. The 1st type of perturbation is simply a perturbation of a mode by the presence of another core, *i.e.* center-core modes are perturbed by the helical side core and side-core modes are perturbed by the central core. This causes coupling between the modes with the same circular polarization, *i.e.* RCP to RCP and LCP to LCP. In the above formula Eq. (13), this corresponds to $\Delta m$ values with $\Delta s$ = 0. The 2nd type of perturbation arises from local linear birefringence created in the spacing between the two cores due to the difference in the thermal expansion between the two cores and the cladding during the fiber draw process. This perturbation causes coupling between orthogonally polarized modes, *i.e.* RCP to LCP and vice versa. In Eq. (13), this corresponds to $\Delta m$ values with $\Delta s$ = −2 and + 2. The 3rd type of perturbation is associated with the local “shear” birefringence created in the spacing between the two cores due to viscosity of the glass flow during the spinning of a heat-softened fiber perform while drawing CCC fiber. This perturbation essentially acts as optical activity, causing mixing between longitudinal and transverse modal field components (similar to the case of a twisted cylindrically-symmetric conventional fiber [23]) and is described by $\Delta m$ values with $\Delta s$ = −1 and + 1 in Eq. (13).

Direct experimental demonstration and measurement of inter-core modal interactions involving modal angular momentum are presented in Fig. 3. We explored CCC fiber performance in the vicinity of low-loss 1030nm resonance marked in Fig. 2b by line C-D. At this wavelength central-core fundamental LP_{01} and side-core LP_{11} are quasi-phase-matched, corresponding to ${l}_{1}$ = 0, ${l}_{2}$ = −1, $\Delta s$ = −2 with the resulting $\Delta m$ = 0 - 1 - 2 = - 3, which can be easily verified by plugging the values of these numbers and propagation constants at 1030nm into Eq. (13). Figure 3a and Fig. 3b show the output from the same 1.5m-long CCC fiber sample when only the central core is excited: one is measured at wavelengths outside the 1030nm resonance and the other is measured at the 1030nm resonance. One can see that outside the resonance output signal is observed only in the central-core, while at the resonance fiber output signal is observed in both the central core and side-core, thus directly confirming inter-core modal interaction at 1030nm. Furthermore, central core output appears to be single-mode (as confirmed by more rigorous measurement described below), while side-core mode, observed at the resonance, exhibits a clear ring-like higher-order mode pattern, consistent with LP_{11} optical vortex. The later was further confirmed experimentally: we used a narrow linewidth laser operating at 1030 nm to excite the side core and, in addition, to produce a reference plane-wave, and the overlap of them will produce an interference pattern presented in Fig. 3c, which indeed confirms that the side core output is an optical vortex with the topological charge |1| plotted in Fig. 1d [22].

## 6. Effective single mode operation in CCC fibers

One remarkable and practically important effect that this QPM coupling in a CCC structure enables to achieve is single transverse mode propagation in a large diameter core fiber, at core sizes well beyond conventional single-mode fiber limit [20]. This has already been shown in Fig. 2d, where at wavelengths longer than ~950nm fundamental mode is transmitted with negligible loss (except within narrow-band resonances), while transmission of all higher order modes is suppressed by at least three orders of magnitude. In fact, this CCC fiber central core, which would by itself support five transverse spatial modes, becomes effectively single-mode in this domain, with performance indistinguishable from truly single-mode fiber. Qualitative difference between effectively-single-mode large core CCC fibers and conventional large mode area (LMA) fibers is clearly revealed in Fig. 4 . Spectral measurements have been performed by sending a broad-band amplified-spontaneous-emission source operating in the 1010nm-1090nm spectral region into the fiber under test and obtaining the spectrum with an optical spectrum analyzer through a standard single mode fiber acting as a spatial filter [24]. Here Fig. 4a shows the transmission spectra of a conventional multi-mode core (30µm diameter) fiber measured at different transversal beam-launching positions with respect to fiber axis, which shows prominent spectral modulation indicating the multimode nature of this LMA fiber. This spectral modulation (which can also be referred to as spectral “beating”) is a result of spatial interference between different transverse modes, which propagate in the test fiber with different group velocities [24]. In other words, there will be no spectral “beating” at all if there is only one mode supported. Figure 4b shows similarly obtained transmission spectra of the same 1.5m-long 35µm central-core CCC fiber sample without a trace of any spectral modulation. This clearly shows that, despite large core size, CCC fiber performs as a truly single-mode fiber, producing fundamental mode output irrespective of fiber excitation conditions.

## 7. Conclusion

In conclusion, we have demonstrated a completely new degree of freedom in tailoring fiber modal properties by using optical-angular-momentum-assisted wave interaction in CCC fiber geometry. As an example we have shown that this new degree of design freedom enables single-mode fibers with very large cores, which are critically important for future high power fiber laser technology. Further ongoing studies of CCC structures indicate that other functionalities can be achieved in CCC geometry, including controlling nonlinear interactions of high-intensity beams and controlling orbital-angular (as well as spin-angular) momentum of a beam for particle trapping and manipulation, quantum communication, and quantum computing and information encoding in multi-dimensional quantum space.

## Acknowledgments

This work was supported by US Army Research Office grant W911NF051057.

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