Angular momentum of optical modes in a silicon channel waveguide

The spin and orbital angular momentum (OAM) of optical ﬁelds in a silicon channel waveguide are investigated. A vortex beam carrying OAM can be generated by a superposition of two quasi-TE modes. The longitudinal OAM of the superposed vortex beam is intrinsic and mediated by the superposition coefﬁcient. Due to high transverse conﬁnement of the silicon waveguide, the polarization and OAM are inevitably coupled. Their hybridization degree is well described by joint average transverse wave vectors. Further, the OAM part can be separated using the topological Pancharatnam charge and Stokes parameters. Combining polarization and OAM indices, we ﬁnd that the whole space of structure parameters of the waveguide can be divided into three regimes. Our result offers insight into the relationship between angular momentum and mode conﬁnement, which is beneﬁcial for potential applications of vector beams in photonic integrated circuits.


I. INTRODUCTION
Light possesses a set of dynamical characteristics of energy, momentum, and angular momentum (AM) [1].The AM can be separated into spin and orbital contributions due to the vectorial nature of the electromagnetic fields [2,3].The spin angular momentum (SAM) is related to the intrinsic polarization degrees of freedom.The spatial distribution and propagation of the optical fields determine the orbital angular momentum (OAM).Stemming from the internal coupling properties of Maxwell's equations, the interaction between the polarization and the orbital degrees of freedom happens naturally [4].This spin-orbit interaction of light has a fundamental importance in optical processes and explains many striking phenomena involving the areas of nano-optics, plasmonics, nonlinear optics, and quantum informatics [5][6][7][8][9][10][11].
Over the past decades, the SAM and OAM of light have been well studied for free-space electromagnetic fields, even nonparaxial [12][13][14].Recently, these concepts have entered the light-wave guiding systems (e.g., optical fibers) arising from the important application for information processing [15][16][17].The guiding systems intimately relate to inhomogeneous optical media, where structured lights are supported.An important category of guiding systems is optical fiber, wherein the angular momentum has been studied by many researchers [17][18][19][20].Most optical fibers are weakly guiding systems and propagation of lights approaches paraxial, resulting in decoupling between the spin and the orbital degrees of freedom.However, the spin-orbit coupling inevitably appears under the consideration of transverse confinement and removes the independence of SAM and OAM.Now it is well known that the guided light beams propagating in an inhomogeneous media possess mutually coupled SAM and OAM [21][22][23].As a result, the polarization and the propagation of optical beams are connected via the spin-orbit interaction of light in the guiding systems.In particular, the concepts of transverse spin and spin-momentum locking in evanescent waves (associated with the guided waves) have been proposed to achieve spin-controlled directional guiding of light [24][25][26].
More recently, another important guiding system-the silicon waveguide-has attracted great attention due to the pursuits of robustness, miniaturization, and energy efficiency of photonic integrated circuits, where the silicon waveguide is the basic unit and component [27,28].Because of the highly confined modes in silicon waveguides, the SAM and OAM are strongly coupled.The spin-orbit interaction of light in silicon waveguides promises to be a powerful tool available for fabricating integrated optical devices with special functionality such as polarization rotation, directional guiding, chiral sensing, and SAM-to-OAM transformation [29][30][31][32][33][34][35][36].Due to the planar technology, the silicon channel waveguide is the most common structure and compatible with current technologies of integrated optical circuits.It is important to note that vortex modes carrying OAM are naturally supported in cylindrically symmetric waveguides (e.g., optical fibers).Apparently, the rotation invariant is no longer present in channel waveguides.Therefore, compared with the typical weak-guide fibers, the silicon channel waveguides are strongly confined optical structures without cylindrical symmetry.This distinct difference may bring special properties of angular momentum and hold great promise for studying many intriguing phenomena such as the spin-Hall effect of light, extraordinary optical momentum, spin-orbit hybrid entangled states, and quantumoptical analogies [37][38][39][40][41][42].
In our previous work, we proposed that a first-order OAM field can be generated in a Si/SiO 2 channel waveguide with the superposition method [33].In that paper, two quasi-transverse electric eigenmodes, TE 01 and TE 10 , were superposed to generate first-order OAM fields.Meanwhile, we designed a coupling structure as an OAM generator with the excitation of fundamental modes and calculated the transformation efficiency (i.e., purity) of the OAM charges by the expansion method.It should be noted that the calculation of the charge purity was based on the common assumption that the charge of the OAM basis (perfect superposition) itself is ±1, which corresponds to the mode azimuthal order (i.e., one node in the transverse field distribution) of TE 01 /TE 10 (see the Appendix for details).However, note that there is no rotationinvariant symmetry in the silicon channel waveguide and its modes are highly confined due to the large refractive index difference.These conditions are different from free-space and cylindrical waveguides and make the assumption unreliable.In this regard, it is necessary to reexamine the OAM value of generated modes in such a channel waveguide to shed light on the distinctiveness.
In this work, we investigate systematically the angular momentum behaviors of superposed modes of TE 01 and TE 10 in a Si/SiO 2 channel waveguide.The modulation of the superposition coefficient on OAM and SAM is demonstrated.The hybridization of OAM and SAM, induced by the high transverse confinement, is analyzed in the spatial frequency domain.In particular, we find that the separation of OAM density can be well described by topological Pancharatnam charge and Stokes parameters.This paper is organized as follows.In Sec.II, first we introduce the formalism for angular momenta in inhomogeneous media.Then we demonstrate the basic feature of angular momenta in a silicon channel waveguide.And the topological Pancharatnam charge is depicted for characterizing helical phases.Section III shows the maximum achievable OAM charge in a silicon channel waveguide and unfolds the hybridization of OAM and SAM via three special cases.We develop the average transverse wave vector for the description of confinement and build the correspondence with the hybridization of OAM and SAM.Separation of the OAM density is achieved with the use of topological Pancharatnam charge and Stokes parameters.Finally, we present the conclusion in Sec.IV.

A. Formalism for the angular momentum
To tackle angular momenta in an inhomogeneous medium, we introduce an effective formalism describing the canonical momentum, spin, and orbital AM of light in isotropic dispersive media, which is developed and examined recently in cylindrical dielectric fibers and metallic wires [19,43].First, a cycle-integral energy density in such a medium can be described by the Brillouin expression [44] where E(r) and H(r) are the complex electric and magnetic field amplitudes, and {ε, μ} = {ε, μ} + ωd{ε, μ}/dω are the dispersion-modified absolute permittivity ε(r, ω) and permeability μ(r, ω), with ω the angular frequency of light.Then the canonical momentum, spin, and orbital AM densities are expressed as [19,43] L = r × P; here we have used the notation As demonstrated in Ref. [19], for cylindrical waveguides, considering a guided mode with propagation constant β along the z direction, the longitudinal momentum and energy fulfill the relation P z / W = β/ω, where . . .denotes the integration over the transverse (x-y) plane.

B. Angular momentum of superposed modes in silicon channel waveguides
Similarly to the structure parameters of the Si/SiO 2 channel waveguide in Ref. [33], the silicon core with a cross section of 720 nm × 600 nm is embedded in silicon dioxide.The corresponding refractive indices are set at n Si = 3.45 and n SiO 2 = 1.46.For simplicity, only nondispersive and lossless media are considered in what follows, thus {ε, μ} = {ε, μ}.It can be calculated that TE 01 and TE 10 are degenerate at wavelength λ = 1550 nm with propagation constant k z = 10.48 rad/μm (see the Appendix for details).As shown in Fig. 1(a), a vortex beam can be generated by the superposition of TE 01 and TE 10 with a phase shift of π/2.Without loss of generality, the generic superposed field can be expressed in a normalized form, + αe iδϕ E (10) ] where E (01) and E (10) are power-normalized electric fields of TE 01 and TE 10 , respectively, and {α, δϕ} denotes the superposition coefficient with the amplitude weight (0 α 1) and phase difference (0 δϕ < 2π ).Using Eqs. ( 1) and (2), we calculate average charges of the canonical OAM (ω L z / W ) and SAM (ω S z / W ) in relation to {α, δϕ}, which are shown in Fig. 1(b).Apparently, the average OAM charge is mediated by {α, δϕ}, which is analogous to that of the Laguerre-Gaussian mode LG 01 generated by Hermite-Gaussian modes HG 01 and HG 10 [2].However, the tunable behavior of the SAM of E sp is different from that of LG 01 (which is always 0).This occurs because there are no pure TE modes in channel waveguides, so different field components will mix and generate elliptical polarization states in the superposed fields [see Fig. 1(a)].To assess the intrinsic attribute of OAM in a channel waveguide, we perform relevant calculations for maximally twisted E sp at {α, δϕ} = {0.5, π/2}, which is the scenario exactly demonstrated in Fig. 1(a).The calculated distribution of the longitudinal OAM density (L z ) on the waveguide cross section (x-y plane) is displayed in Fig. 1(c).We examine two cases: no displacement shift (origin of r ⊥ coincides with beam center) and a shift of (100 nm, 50 nm) for the origin.The calculations give that the cross-section integral ω L z / W equals 0.87 in both cases, though their density distributions are different.This characteristic indicates that the generated OAM field has an intrinsic longitudinal orbital angular momentum [25,45].The result that the net transverse momentum is exactly 0 for a guided mode can be easily understood according to the interpretation given in Ref. [45].It can be examined that the relation P z / W = k z /ω is also fulfilled in such a channel waveguide.Certainly, the transverse OAM is still extrinsic [25].Note that the average canonical OAM charge is not exactly equal to 1 as the assumption in our previous paper [33].The assumption of discrete topological charge is inherited from that of a spiral phase front in scalar vortices, where the phase varies as integral multiples of 2π along a circular path.But this assumption is elusive for vectorial fields, such as the superposed modes in a silicon channel waveguide.In fact, for vector vortices, the discrete charge can reappear upon the introduction of a topological Pancharatnam charge (TPC).

C. Topological Pancharatnam charge
Here, using the Pancharatnam connection, the phase for any field |a(x, y) = [E x (x, y), E y (x, y)] T can be written as where the left circularly polarized field |e + = (1/ √ 2)[1, i] T is used as the reference field.Similarly to scalar vortex beams, we can define the TPC for vector vortices as [46][47][48] where φ is the angular coordinate in the transverse plane, l TPC = ∂ψ P+ /∂φ is the density of C TP , and C is a circular path around the beam center.
With the phase definition in Eq. ( 4), the phase fronts for the fields in Fig. 1(a) are calculated and displayed in Fig. 2(a).To demonstrate more clearly, Fig. 2(b) shows the φ dependence of ψ P± along the path of the cyan circle shown in Figs.1(a) and 2(a).It is easy to read that C TP equals 1 and −1 for modes TE 01 and TE 10 , respectively.The magnitude of C TP is determined by the mode azimuthal order (node number) stemming from topological protection [46].Then the C TP of their superposition field E sp also can take a value of 1 or −1 depending on {α, δϕ}.In Fig. 2, C TP = 1 for E sp under the condition of {α, δϕ} = {0.5, π/2}.Note that all superposed fields have the same TPC magnitude (i.e., |C TP | = 1), but the TPC densities (l TPC ) are different, which can be evaluated by the slope (∂ψ P+ /∂φ) of the curves in Fig. 2(b).For TE 01 and TE 10 , l TPC is varied along φ and the maximal magnitudes of l TPC are reached at the nodes of the fields.While for E sp at {α, δϕ} = {0.5, π/2}, l TPC is smoother and undulates slightly around 1 versus φ.
Figure 1(b) shows that the average canonical OAM charge can be tuned from −0.87 to 0.87, which means that the maximal magnitude is 0.87.It clearly shows that the maximum achievable OAM charge (equal to 0.87) is not equal to the TPC (equal to 1).This indicates that the topological Pancharatnam charge is incomplete to characterize the OAM charge for such a vector vortex.So one question to ask is how transverse confinement determines the maximal magnitude of the average canonical OAM charge in a silicon channel waveguide.Moreover, is it possible to reach 1, which exactly equals the mode azimuthal order (i.e., C TP ) of TE 01 /TE 10 ?To better appreciate these relations, we make a systematic study of the OAM and SAM in a silicon channel waveguide via sweeping structure parameters and reveal their dependencies in the following section.

III. THE MAXIMUM ACHIEVABLE OAM CHARGE A. AM map in the structure-parameter space
For such a simple structure as the Si/SiO 2 channel waveguide, there are two parameters, silicon-core width w Si and height h Si [see Fig. 1 1) and (2).For each set {w Si , h Si }, the maximum charges can be filtered and are plotted in Figs.3(a) and 3(b) for the OAM and SAM parts, respectively.It is noteworthy that, generally, the superposed vortex beams are not stable due to the different propagation constants of TE 01 and TE 10 , however, in this work we would like to explore the maximum achievable OAM charge for superposition of TE 01 and TE 10 in the channel waveguide.Of course, a special case is that a stable vortex beam can be realized with a degenerate TE 01 and TE 10 , which is shown by the dashed red curve in Fig. 3.For brevity in the following text, the maximum achievable charges of canonical OAM and SAM are referred to as the OAM and SAM charges, respectively, unless otherwise noted.From Fig. 3(a), on the whole, it can be found that the OAM charge increases as the width (w Si ) and aspect ratio (w Si /h Si = [h Si /w Si ] −1 ) of the silicon core increase.In contrast, the SAM charge is inversely proportional to the two factors as shown in Fig. 3(b).The larger w Si means weaker confinement of modes, which trends to the paraxial limit.The confinement leads to hybrid eigenmodes, which possess a space-variant state of polarization (SOP).When w Si /h Si approaches 1, the overlap between the TE 01 and TE 10 fields becomes large and causes strong mixing of the SOP in the superposition.In general, an intense space-variant SOP promotes the SAM charge and degrades the OAM charge, which can be supported phenomenologically by comparing Figs.3(a) and 3(b).In the following subsections, we further reveal the specific relation between OAM and confinement by investigating three special cases-the degenerate condition, a varying aspect ratio with a fixed height, h Si = 600 nm, and an OAM charge equal to 1-which are shown by three different curves in Fig. 3.

B. Average transverse wave vectors
To characterize the degree of mode confinement, a transverse wave vector is introduced in the spatial frequency domain.In order to obtain the transverse wave vector, we first perform Fourier transforms on E x = Re[E x (x, y)] and E y = Re[E y (x, y)] and generate two-dimensional maps F (E x ) and F (E y ) as a function of the transverse wave vector k ⊥ = (k x , k y ), respectively.Then the wave vectors corresponding to the maximal magnitudes of F (E x ) and F (E y ) read as furthermore, the two wave vectors are weighted by their electric components and the average transverse wave vector is expressed as finally, to eliminate confusion of the arbitrary magnitude of the electric intensity, k⊥ is constrained by k2 We can find that k⊥ contains the information, given by Eqs. ( 6) and ( 7), of both the transverse confinement and proportion of (E x , E y ).Obviously, the paraxial limit corresponds to | k⊥ | approaching 0. For each set {w Si , h Si }, we repeat the above procedure and obtain k(01) ⊥ and k(10) ⊥ for TE 01 and TE 10 , respectively, which are illustrated in Fig. 4(a) for {w Si , h Si } = {720 nm, 600 nm}.For TE 01 , a node of E x (the dominant electric component) appears in the y direction, which shows a higher confinement along the y direction and generally results in kx < ky for k(01) ⊥ .In the limit of a perfectly pure TE 01 , kx tends to 0 arising from E y = 0 and the tiny confinement along the x direction.On the contrary, kx > ky for k(10) ⊥ and ky approaches 0 in the perfectly pure limit.Thus, in the k⊥ space plot, k(01) ⊥ and k(10) ⊥ are located in the top-left half and bottom-right half, respectively, which are shown in Fig. 4

(b). With the two wave vectors k(01)
⊥ and k(10) ⊥ , the opening angle between their directions and the median of their direction angles are given by ⊥ is also a composite index, which manifests itself in complementarity in the distributions of both the SOP (vectorial property) and the electric intensity.

C. Superposition with degenerate modes
According to the structure parameters of the degenerate curve shown in Fig. 3, the degenerate propagation constant of TE 01 /TE 10 and the OAM charge of their superposed fields as a function of w Si are calculated and displayed in Fig. 5(a).The smallest cross section of the silicon core for no-cutoff degenerate TE 01 /TE 10 is exactly square, where the degenerate modes have the smallest propagation constant due to the highest mode confinement.Their superposition generates a cylindrical vector beam, whose OAM and SAM charges are both 0 [see Fig. 5(a)].Growth of w Si , which means a reduction in the mode confinement, can increase the propagation constant and the OAM charge.In the paraxial limit, the OAM charge is close to 1 and the SAM charge approaches 0.
For the degenerate case in k⊥ space, ξ o ⊥ gradually increases from 0 to π/2 following w Si [see Fig. 5(b)], which denotes that the joint-SOP purity of TE 01 -TE 10 is increased with reduced confinement.But ξ m ⊥ goes down from π/4 to a certain value and increases close to π/4 again in the paraxial limit.The maximum for the ξ m ⊥ deflection (|ξ m ⊥ − π/4|) means a low complementarity between TE 01 and TE 10 .This maximal ξ m ⊥ deflection, associated with a poor joint-SOP purity (ξ o ⊥ < π/2), induces a large mixture of SOPs in a superposed field and corresponds to the maximum of the SAM charge, which is illustrated by comparing ω S z / W in Fig. 5(a) with ξ m ⊥ in Fig. 5(b).Note that, in the paraxial limit under the degenerate condition, the OAM charge tends to 1 but is always smaller than 1, which results from the relatively high complementarity between TE 01 and TE 10 .A further explanation is given in the following cases.The degenerate case well demonstrates that the average transverse wave vectors k(01) ⊥ and k(10) ⊥ can offer rich information about mode confinement, which is useful to evaluate the behaviors of the SAM and OAM of superposed fields.To further manifest this, we study a generic case of varying w Si at h Si = 600 nm.

D. Varying w Si at h Si = 600 nm
Figure 6(a) displays the w Si dependence of the SAM and OAM charges in this case.Following the growth of w Si , the SAM charge increases from 0 to a certain value and decreases to 0 again, which is similar to that in the degenerate case.But the w Si dependence of the OAM charge is different from that in the degenerate case; the OAM charge starts at 0, increases continually, and finally moves beyond the value of 1.An OAM charge larger than 1 means that it can break through the mode azimuthal order of TE 01 and TE 10 (i.e., the OAM charge is larger than |C TP |).In the spatial frequency domain, the related average transverse wave vectors k(01) ⊥ and k(10) ⊥ are displayed in Fig. 6

(b). The decreasing magnitudes of k(01)
⊥ and k(10) ⊥ with the growth of w Si express the mode confinements becoming weaker.Concomitantly, ξ o ⊥ gradually grows from 0 to a constant but not π/2, and ξ m ⊥ decreases from π/4 to a certain value, goes back to nearly π/4, then decreases again.⊥ − π/4|, which resembles their dependence in the degenerate case.Another scenario is that the OAM charge takes a value of 1, which corresponds to the return of ξ m ⊥ π/4.Across this setting, the OAM charge increases continually with the increment in |ξ m ⊥ − π/4|.To provide deeper insight into the origin of the OAM, we separate the OAM density into two contributions with the approach introduced in Refs.[47,48].For any field |a , the normalized Stokes vector S = (S 1 , S 2 , S 3 ) T can be defined by S j = a|σ j |a /I E ( j = 1, 2, 3), where σ j are the Pauli matrices and I E = a|a is the electric intensity.Then the Poincaré sphere can be constructed by mapping the Stokes vector S on Cartesian coordinates, whose azimuth (ψ S ) is calculated by According to Refs.[47,48], the longitudinal OAM density j o z can be expressed as In Eq. ( 11), the first term exactly corresponds to the density of the TPC given in Eq. ( 5) and the second term represents the contribution induced by a spatially varied SOP (termed the SOP-OAM density and denoted l SOP ).With the ratio of angular momentum to energy that was examined by Allen [2], the average OAM charge can be calculated using Eq. ( 11) as where r is the radial coordinate in the transverse plane, the first term is the corresponding average TPC-OAM charge, and the second term is the average SOP-OAM charge.With the use of Eq. ( 12), the calculated TPC-and SOP-OAM charges are plotted in Fig. 7(a) for the case of h Si = 600 nm.It clearly shows that the sum of the TPC-and SOP-OAM charges matches well with the calculation for the average canonical OAM charge using Eq. ( 2).According to the two special settings of w Si in Fig. 6, the charts of OAM charge contributions can be roughly divided into three stages as shown in Fig. 7(a).The first stage, at the beginning of w Si , shows that both the TPC-and the SOP-OAM charges vary strongly but offer opposite contributions.In particular, at the lowest setting, w Si = 600 nm, corresponding to a square cross section, the TPC-and SOP-OAM charges are equal but opposite, which induces a total OAM charge equaling 0. In the second stage, the variation of the TPC-OAM charge is moderated and the SOP-OAM charge approaches 0. In the third stage, the SOP-OAM charge remains 0, while the TPC-OAM charge increases constantly from 1. Comparing with Fig. 6, we find that the SOP-OAM evolves strongly in the first and second stages, where both the poor joint-SOP purity (ξ o ⊥ < π/2) and the degradation of complementarity (mainly manifested in the SOP distributions) occur in tightly confined fields.In the third stage, the degradation of complementarity (mainly manifested in the distributions of the transverse electric intensity) appears again due to the high aspect ratio of the waveguide and leads to an increment in the TPC-OAM charge.
To further demonstrate their dependencies, the transverse electric intensity, TPC-OAM density, and SOP-OAM density are calculated by Eq. ( 11) and displayed in Figs.7(b)-7(e) for four settings of w Si .As shown in the top panels in Figs.7(b)-7(e), the SOP purities are enhanced and the aspect ratio of the intensity distribution increases following an increment in w Si .These indicate that, with regard to complementary symmetry about the x = y (i.e., 45 • ) line, the reflection on the intensity distribution is reduced while the similarity of the SOP distribution is roughly promoted.In the middle and bottom panels in Figs.7(b)-7(e), we focus on the distributions within the dashed black boxes since they provide the major contributions to the weighted average OAM charges.The SOP-OAM density decreases and becomes 0 with the growth of w Si , which is shown in the bottom panels in Figs.7(b)-7(e).But for the TPC-OAM density, some local densities are very high at small w Si [see middle panel in Fig. 7(b)], and the density distribution becomes smooth at large w Si [Figs.7(c) and 7(d)], then some local densities increase to very high values again at a larger w Si [Fig.7(e)].In high confinement (i.e., with a small w Si ), the SOP-OAM contribution counteracting the TPC-OAM contribution makes the total OAM charge smaller than 1.In weak confinement, the total OAM charge can achieve a value larger than 1 since the SOP-OAM density becomes almost 0.This result shows that the OAM charge can reach beyond the mode azimuthal order of TE 01 and TE 10 .In the case of h Si = 600 nm, we can find that the ξ m ⊥ deflection (i.e., degradation of complementarity) induces a large SOP-OAM contribution in high confinement, while it almost wholly converts to the TPC-OAM contribution in weak confinement.In particular, the TPC-OAM charge arising from the degradation of complementarity can promote the total OAM charge beyond the mode azimuthal order.Due to the particularity and typicality of an OAM charge equal to 1, we show the behaviors under this condition in the next subsection.

E. Realizing ω L z / W = 1
With the help of the contour map of OAM charge shown in Fig. 3(a), it is easy to extract the isoline of ω L z / W = 1 all the parameters except for the deeply confined fields at an initially small w Si .This means that a small ξ m ⊥ deflection with a not-so-good joint-SOP purity occurs at very high mode confinement, where the SOP is mixed slightly and the SOP-OAM contribution is involved in the total OAM charge.The corresponding separated OAM contributions are calculated and displayed in Fig. 8(b).As expected, apart from the beginning of w Si , the TPC-OAM and SOP-OAM charges remain almost-constant.At the beginning of w Si , the TPC-OAM and SOP-OAM charges show some small shifting but they can compensate each other to keep the total OAM charge value 1.And the expected small rise in the SAM charge also can be found at the beginning.To further illustrate, the related density distributions are demonstrated for two settings of w Si in Figs.8(c) and 8(d).For a low setting, w Si = 800 nm, some local SOP-OAM densities are not equal to 0 [see bottom panel in Fig. 8(c)].But this will disappear in the larger setting of w Si = 3000 nm [Fig.8(d)].However, the TPC-OAM density is not smooth in distributions and can reach values larger than 1 for both settings [see middle panel in Figs.8(c) and 8(d)].But the local TPC-OAM density with a large amount appears farther away from the region of high electric intensity at larger w Si than at small w Si .In general, there is a series of options of structure parameters to generate an OAM charge equaling 1 and these options almost represent the highest complementarity between TE 01 and TE 10 in the whole parameter space.

F. The structure-parameter space is divided into three regimes
Based on the preceding demonstrations, we can divide the full parameter space into three regimes with two curvesω L z / W = 1 and max[s(w Si )] with s(w Si ) = ω S z / Was a function of w Si , which are plotted in Fig. 3.The curve of max[s(w Si )] means that the degree of SOP mixing is maximal in superposed fields, which also implies that the ξ m ⊥ deflection reaches the extremum.Before this extremum (lower side of w Si ), both the TPC-and the SOP-OAM charges vary strongly and involve deeply in the contribution to OAM, but their sum (i.e., the total OAM charge) stays at a low value (<0.8).In the upper end of this extremum, the variations of the TPCand SOP-OAM charges become moderate and the SOP-OAM charge approaches 0 following the growth of w Si .Then reaching the curve of ω L z / W = 1, ξ m ⊥ returns to almost π/4, which denotes a high complementarity between TE 01 and TE 10 .Further beyond ω L z / W = 1, ξ m ⊥ moves away from π/4 again due to the degradation of complementarity in the electric intensity distributions.In this regime, the total OAM charge (>1) grows constantly, resulting from the increment in the TPC-OAM charge, whereas the SOP-OAM charge remains almost 0. In general, the paraxial limit can be achieved by increasing w Si , where the three curves of max[s(w Si )], degenerate condition, and ω L z / W = 1 will merge in this limit.

IV. CONCLUSION
This work has studied the SAM and OAM of highly confined vertex fields in silicon channel waveguides, which are generated by the superposition of TE 01 and TE 10 modes.Compared with cylindrical fibers, there are similar features for the SAM and the intrinsic longitudinal OAM: In high confinement, the space-variant SOP modes induce a nonzero SAM charge and degrade the TPC-OAM charge; in the paraxial limit, the pure SOP modes give a zero SAM charge and promise that the total OAM charge is only contributed by the TPC-OAM.However, there are also several significant differences: (i) In cylindrical fibers, there are eigenmodes that carry OAM, even though they are not easily excited by general optical sources.The reason is that the rotation invariant guarantees the support of spatially twisted light.In channel waveguides, there is no rotation invariant, so a stable twisted light cannot be supported in principle.But the x-and y-mirror symmetries allow the channel waveguide to support the vertex field upon the occurrence of accidental degeneracy (e.g., degeneracy between TE 01 and TE 10 ).This requirement is critical and ideal since the stability of the vortex greatly depends on the wavelength of light and structure parameters.Under the degenerate condition, the relation P z / W = β/ω also holds for superposed modes in channel waveguides.In general conditions, the OAM charge of vortex fields cannot be preserved during propagation owing to the different propagation constants of TE 01 and TE 10 .The spatial beating generates a periodically varied OAM charge, whose "amplitude" is the maximum achievable OAM charge investigated in this work.
(ii) In cylindrical fibers, the AM densities of twisted eigenmodes are uniform along the φ direction due to the rotation invariant, which makes the total AM charge exactly equal to the mode azimuthal order (equivalent to C TP ).The difference between high and low confinements is just the different proportions between SAM and OAM.In channel waveguides, the distributions of AM densities just satisfy the reflection symmetry, which implies that the uniformity of density along the φ direction is not necessary.This means that the total AM charge can break through the restriction of C TP .The total AM charge can be smaller and even larger than C TP .
(iii) For the simplest step-index fibers, there is only one structure parameter, the radius of the core.The radius solely determines the mode confinement, which also means the proportion between SAM and OAM is locked by this radius.For channel waveguides, there are two independent parameters, w Si and h Si , which allow adjustment of the proportion between SAM and OAM by changing the aspect ratio w Si /h Si with a comparable confinement.
In the separation of the OAM charge, we used the approach proposed in Refs.[47,48].It should be noted that this is just an "empirical" application because the approximations of the paraxial limit are used in Eq. (11).But the calculation results matched well with the results calculated directly by fields using Eq. ( 2).The remarkable application of this approach is that the accurate OAM charge can be measured with two detectable parameters [47,49].Here, we find that it is also helpful to analyze the contributions of OAM and manipulate AM densities in design.
In summary, we have studied the SAM and OAM of light in silicon channel waveguides.A vortex beam carrying OAM can be generated by a superposition of the TE 01 and TE 10 modes.The longitudinal OAM of the superposed vortex beam is intrinsic and mediated by the superposition coefficient.The SAM and OAM of the generated vortex fields are strongly coupled in silicon waveguides due to the transverse confinement.To characterize the confinement, we proposed the average transverse wave vector and saw a good correspondence with the hybridization of SAM and OAM.Combining the SAM and OAM charges, the whole structure parameter space can be divided into three regimes.Meanwhile, separation of the OAM was achieved with the use of the topological Pancharatnam charge and Stokes parameters.The investigation permits increased functionality, applicability, and design freedom of the angular momentum of optical fields in silicon channel waveguides.
In a dielectric slab waveguide, perfect transverse electric (TE) and transverse magnetic (TM) modes are supported, since there is only one direction confinement [50].Typically, assuming that the modes are confined in the y direction and propagate along the z direction, the TE modes have the components {E x , H y , H z }, while the TM modes possess the components {H x , E y , E z }.However, for a channel waveguide, there are only hybrid eigenmodes due to confinements from both the x and the y directions.This means that all six components are possessed by all eigenmodes.To classify these modes, we assume that the quasi-TE modes are the mode fields that have the major components {E x , H y , H z }, and the quasi-TM modes are the mode fields that have the major components {H x , E y , E z }.That is, viewed from the transverse electric components (i.e., E x and E y ), the eigenmodes with major E x are denoted quasi-TE modes and the eigenmodes with major E y are denoted quasi-TM modes, which are usually called TE and TM modes, respectively, for short.Further classification is performed by numbering the FIG. 9. Structural dispersion curve for a silicon channel waveguide for a fixed height of 600 nm at a wavelength of 1500 nm.The insets at the right schematically show the amplitude distributions of E x and E y for the lowest six modes.
nodes along the x and y directions, which is on E x for TE modes and E y for TM modes.Figure 9 displays the lowest six eigenmodes for a silicon channel waveguide via numerical calculations [51].Meanwhile, each mode is labeled with the described classification approach.In the calculations, we sweep w Si around 720 nm with a fixed h Si = 600 nm at a wavelength of 1550 nm.It is found that the TE 01 and TE 10 modes are degenerate at w Si = 720 nm.

FIG. 1 .
FIG. 1.(a) Schematic for generation of a vortex beam with modes TE 01 and TE 10 in a Si/SiO 2 channel waveguide.The magnitude and polarization state for the transverse component of electric fields are expressed by color map and white curves, respectively.The dashed cyan circles mostly cross the magnitude peaks of both TE 01 and TE 10 .The dashed black boxes show the boundaries of the silicon-core region, whose size is {w Si , h Si } = {720 nm, 600 nm}.(b) Dependence of the average canonical OAM and SAM charges of superposed fields on the amplitude weight α and phase difference δϕ for the same structure parameters as in (a).(c) For a superposed vortex field of {α, δϕ} = {0.5, π/2}, distributions of the longitudinal OAM density on the waveguide cross section are calculated and demonstrated without and with shifts of the coordinate origin in the left and right panels, respectively.Black dots point to the beam center and green crosses show the coordinate origin of r ⊥ .The origin shift in the right panel is (100 nm, 50 nm).Both densities are normalized with the maximum of the no-shift case.

FIG. 2 .
FIG. 2. (a) The phase front ψ P+ based on the Pancharatnam connection with the reference field |e + for the corresponding fields displayed in Fig. 1(a).The dashed black boxes and dashed cyan circles are identical to those in Fig. 1(a).(b) ψ P+ along the path of the dashed cyan circle for each mode in (a) versus the angular coordinate φ (see the inset).The slopes of these curves express the density (l TPC ) of the topological Pancharatnam charge C TP .
(a)], for tailoring eigenmodes.The results for {w Si , h Si } dependence of the maximum of the average canonical OAM and SAM charges are demonstrated straightforwardly in Fig. 3. To obtain the results shown in Fig. 3, we first sweep {w Si , h Si } and get the corresponding modes of TE 01 and TE 10 .Owing to the equivalent roles of w Si and h Si and consideration of TE modes, we just sweep the condition h Si w Si .Then E sp can be superposed for different {α, δϕ} by Eq. (3) and used for calculating the average canonical OAM and SAM charges with Eqs. (

FIG. 3 .
FIG.3.Dependence of the maximum achievable charges of the (a) OAM and (b) SAM on the waveguide structure parameters w Si and h Si .In the bottom-left corner of the graphs, the data cannot be accessed due to the cutoff of TE 01 .We set s(w Si ) = ω S z / W as a function of w Si and plot max[s(w Si )] versus h Si with the light-blue dash-dot-dotted line.The dashed red curve expresses the degenerate condition of TE 01 and TE 10 , whose propagation constants are the same for structure parameters picked from this curve.The dotted green curve denotes a line of structure parameters with a fixed height h Si = 600 nm.The dash-dotted black curve is a contour line for which the OAM charge equals 1.

FIG. 4 .
FIG. 4. Average transverse wave vectors of TE 01 and TE 10 .(a) Fourier transform is performed on each component of the transverse electric fields of TE 01 and TE 10 .Using the calculated 2D maps in the spatial frequency domain, the transverse wave vectors (k xF ⊥ and k yF ⊥ ) corresponding to the maximum can be extracted.(b) The average transverse wave vectors ( k(01) ⊥ / k(10) ⊥ ) of TE 01 /TE 10 are calculated and depicted in k⊥ space, where the related opening angle (ξ o ⊥ ) and angle of the median (ξ m ⊥ ) are also shown.
ξ m ⊥ = (arg( k(01) ⊥ ) + arg( k(10) ⊥ ))/2.(9) It is found that ξ o ⊥ denotes the joint influence of transverse confinement on both TE 01 and TE 10 .The magnitude of ξ o ⊥ is a manifestation of the joint-SOP purity of TE 01 -TE 10 in superposition.The larger ξ o ⊥ produces the better joint-SOP purity due to the large separation of k(01) ⊥ and k(10) ⊥ in k⊥ space.As shown in Fig. 4(b), the median of the direction angles of k(01) ⊥ and k(10) ⊥ acts as a reflection (mirror-symmetry) plane for k(01) ⊥ and k(10) ⊥ in k⊥ space, so ξ m ⊥ can provide the information on mode complementarity between TE 01 and TE 10 .The degree of complementarity can be evaluated by comparing ξ m ⊥ to π/4 (corresponding to the critical line of kx = ky ).When ξ m ⊥ = π/4, there is a high complementarity in transverse the electric fields of TE 01 and TE 10 , which means that their superposition can offer a smooth field distribution along the φ direction.The deflection of ξ m ⊥ to π/4 shows the degradation of the smooth superposed field.Note that ξ m

FIG. 5 .
FIG. 5. AM charges and spatial frequencies of modes under the degenerate condition.(a) Silicon-core width dependence of the propagation constant and the AM charges.The guiding line of the structure parameter is redrawn as the degenerate curve shown in Fig. 3(a).(b) Silicon-core width dependence of the average transverse wave numbers, the opening angle, and the angle of the median for k(01) ⊥ and k(10) ⊥ , whose distributions in k⊥ space are displayed in the inset.Markers of the same color indicate calculations with identical structure parameters.For the degenerate condition, k deg z ≡ k (01) z = k (10) z and | kdeg ⊥ | ≡ | k(01) ⊥ | = | k(10) ⊥ |.

FIG. 6 .
FIG. 6. AM charges and spatial frequencies of light fields at h Si = 600 nm.(a) SAM and OAM charges as a function of the silicon-core width.(b) Silicon-core width dependence of the average transverse wave numbers, the opening angle, and the angle of the median for k(01) ⊥ and k(10) ⊥ , whose distributions in k⊥ space are displayed in the inset.Markers of the same color indicate calculations with identical structure parameters.

FIG. 7 .
FIG. 7. OAM charge contributions and densities of vortices at h Si = 600 nm.(a) The average TPC-OAM charge and SOP-OAM charge and their sum as a function of the silicon-core width.Curves for the average canonical SAM and OAM charges are redrawn as those shown in Fig. 6(a).The related density calculations are demonstrated for four silicon-core widths of (b) 600 nm, (c) 700 nm, (d) 1000 nm, and (e) 1500 nm.Each of (b)-(e) has three panels: distributions of the transverse electric field (top), TPC-OAM density (middle), and SOP-OAM density (bottom).The dashed black boxes show the boundaries of the silicon-core region.

FIG. 8 .
FIG. 8. AM charges, spatial frequencies, and charge densities of beams for the scenario of the OAM charge equaling 1.(a) Siliconcore width dependence of the average transverse wave numbers, the opening angle, and the angle of the median for k(01) ⊥ and k(10) ⊥ , whose distributions in k⊥ space are displayed in the inset.Markers of the same color indicate calculations with identical structure parameters.The guiding line of the structure parameter is redrawn as that shown in Fig. 3(a).(b) The average canonical SAM and OAM charges and the average TPC-OAM and SOP-OAM charges (and the sum) as a function of the silicon-core width.The related density calculations are demonstrated for two silicon-core widths: (c) 800 nm and (d) 3000 nm.Each of (c) and (d) has three panels: distributions of the transverse electric field (top), TPC-OAM density (middle), and SOP-OAM density (bottom).The dashed black boxes show the boundaries of the silicon-core region.and derive the structure parameters.With the derived structure relation and associated eigenmodes, we calculate the related parameters in the spatial frequency domain, which are shown in Fig. 8(a).It is found that ξ o ⊥ starting from a certain value finally approaches π/2.And ξ m ⊥ remains π/4 well for almost