Analytic Expressions for Orbital Angular Momentum Modal Crosstalk in a Slightly Elliptical Fiber
AAnalytic Expressions for Orbital AngularMomentum Modal Crosstalk in a SlightlyElliptical Fiber
Ramesh Bhandari
Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, Maryland 20740, [email protected]
Abstract:
Assuming weakly guiding approximation, we examine orbital angular momen-tum (OAM) mode mixing on account of ellipticity in a fiber and derive a complete set ofanalytic expressions for spatial crosstalk, using scalar perturbation theory that incorporatesfully the existing degeneracy between a spatial OAM mode and its degenerate partner char-acterized by a topological charge of opposite sign. These expressions, consequently, includean explicit formula for calculating the 2 π walk-off length over which an input OAM modeconverts into its degenerate partner, and back into itself. We further explore the applicabilityof the derived expressions in the presence of spin-orbit interaction. The expressions consti-tute a useful mathematical tool in the analysis and design of fibers for spatially-multiplexedmode transmissions. Their utility is demonstrated with application to a few mode and amultimode step-index fiber. Index Terms:
Orbital angular momentum, fiber ellipticity, fiber optics, waveguides, modemixing, crosstalk, perturbation theory, analytic expressions, spin orbit interaction
1. Introduction
Currently there exists a vast interest in exploring the possibility of increasing traffic flow multifold on a fibersimply by exploiting the orthogonality of the orbital angular momentum (OAM) modes [1]. However, any sucheffort must take into account fiber imperfections, like ellipticity, which can cause mode-mixing (crosstalk) andreduce the quality of the signal. An analytic treatment of the effect of ellipticity on OAM modes was earlierprovided [2], but it was limited to a low topological charge l , primarily l = l is the OAM carried by asingle photon in units of the reduced Planck’s constant ( h / ( π ) ); besides, the methodology as presented could notincorporate mode mixing with the neighboring OAM modes differing in topological charges by a unit or more.More recently, several authors have carried out numerical studies of the impact of ellipticity on OAM modes ina variety of fibers: ring fiber [3, 5, 7], multi-ring fiber with trench [4], and a graded-index fiber [6] but the studiesare based on computationally-intensive finite-element approximation methods, where it is not always easy toascertain the inherent behavior of the impact of a perturbation like ellipticity in any detailed manner due to theassociated high demands of accuracy.In this work, we assume the weakly-guiding approximation, which permits us to write the vector modes asproducts of the spatial OAM modes and the associated polarization [9, 11]. Further, confining our attention hereto the effect of ellipticity on the spatial mode component only, we employ the scalar wave equation and derive acomplete set of analytic expressions for the spatial OAM modal crosstalk within the framework of perturbationtheory. The obtained expressions include explicit formulas for the 2 π walk-off length, and provide deeper insightinto mode-mixing and thus the crosstalk behavior as a function of ellipticity and the topological charge carriedby the OAM mode. The 2 π walk-off length of a given input OAM mode refers to the distance over which themode transforms itself into its degenerate partner characterized by a topological charge of opposite sign, and backinto itself, under the continuous impact of the perturbation (ellipticity here). We further investigate the effect ofspin-orbit interaction and provide approximate bounds on the applicability of the the derived expressions. Thedeveloped theory is illustrated with application to step-index few mode and multimode fibers.
2. Scalar Wave Equation for the Slightly Elliptical Fiber
We first write the scalar wave equation for the propagation of an OAM mode through an unperturbed fiber as aneigenvalue equation HO l , m ( r , θ ) = β l , m O l , m ( r , θ ) , (1) a r X i v : . [ phy s i c s . op ti c s ] F e b here the Hermitian operator H = (cid:126) ∇ t + k n ( r ) ; k = π / λ , where λ is the wavelength; (cid:126) ∇ t is the transverseLaplacian; n ( r ) = n ( − ∆ f ( r )) ; f ( r ) is the index profile as a function of the radial cylindrical coordinate r only, ∆ = ( n − n ) / ( n ) is the index profile height parameter, with n > n [8]; parameters l and m are respectively thetopological charge (azimuthal mode number) and the radial mode number of the unperturbed OAM mode (denoted OAM l , m ) with an amplitude O l , m ( r , θ ) = (cid:112) N l , m F l , m ( r ) e il θ , (2)where N l , m is the normalization constant; the eigenvalue β l , m is the square of the propagation constant of the OAM l , m mode, which gives the mode its z dependence: ψ l , m ( r , θ , z ) = O l , m e i β l , m z (3)( z axis coincides with the fiber’s axis). The amplitude, O l , m , originates in the reduction of HE l + , m , HE − l − , m , EH l − , m , EH − l + , m ( | l | > ) to O l , m ε + , O − l , m ε − , O l , m ε − , O − l , m ε + , respectively in theweakly guiding approximation ( ∆ <<
1) [9, 10]; ε ± = ( / √ )( ˆ x ± i ˆ y ) are the left (+) and right (-)circular polarizations corresponding to a photon spin + −
1, respectively (in units of h / ( π ) );for the l = HE , m , HE − , m , √ ( T M , m + iT E , m ) , √ ( T M , m − iT E , m ) reduce to the products: O , m ε + , O − , m ε − , O , m ε − , O − , m ε + , respectively [9, 11]. Fig. 1: a) Cross section of the multimode fiber with core radius a and cladding radius b ( >> a ) ; r and θ 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) A slightly elliptical core shown relatively enlarged for clarity; semimajor axis ρ = a ( + ε ) and semiminor axis ρ = a ( − ε ) where ellipticity ε = e / << e is the eccentricity of the ellipse. We model the slightly elliptical fiber (see Fig. 1) as a perfectly round fiber with a modified refractiveindex given by n (cid:48) ( r , θ ) = n ( r ) − ε ∆ n ( ∂ f ( r ) / ∂ r ) r cos ( θ ) ; (4)ellipticity ε ( << ) is defined as the ratio of the difference of the semimajor axis and the semiminor axis to theirsum, and equals e /
4, where e is the eccentricity of the ellipse [2, 8]. Replacing n with n (cid:48) in Eq. 1, the perturbedscalar wave equation is ( H + εδ H ) O (cid:48) l , m ( r , θ ) = β (cid:48) l , m O (cid:48) l , m ( r , θ ) , (5)where O (cid:48) l , m ( r , θ ) and β (cid:48) l , m are the corresponding perturbed mode amplitude and propagation constant, respectively,and δ H = − k n ∆ cos 2 θ ( ∂ f ( r ) / ∂ r ) r . (6)The scalar wave solution incorporating the z dependence is written as O (cid:48) l , m ( r , θ ) e i β (cid:48) l , m z . The O (cid:48) l , m amplitudes, likethe O l , m ’s, form a complete set.The effect of the perturbation εδ H is to cause mixing of an input OAM mode, OAM l , m with other (orthogonal)fiber OAM modes characterized by different pairs of parameters, ( l (cid:48) , m (cid:48) ) (cid:54) = ( l , m ) , as described below.
3. Perturbation Solution
Following [9, 12], we expand the perturbed amplitude O (cid:48) l , m and the perturbed propagation constant β (cid:48) l , m as O (cid:48) l , m = O l , m + (cid:48) ∑ n , k a ( )( l , m )( n , k ) O n , k + (cid:48) ∑ n , k a ( )( l , m )( n , k ) O n , k + ...., (7) (cid:48) l , m = β l , m + β ( ) l , m + β ( ) l , m + ...., (8)where the contributions in different orders of the perturbation parameter ε ( << ) are indicated by the superscriptsin parentheses, and the prime on the summation implies O n , k (cid:54) = O l , m ; all amplitudes are assumed normalized.After inserting the above series in Eq. 5, taking the necessary scalar products to solve in different pertur-bation orders [13–15], we obtain the analytic expressions for the mixing coefficients and the propagation constantcorrections: a ( )( l , m )( n , k ) = εδ H ( n , k )( l , m ) ( β l , m − β n , k ) , (9) a ( )( l , m )( n , k ) = ε (cid:18) ∑ r , s δ H ( n , k )( r , s ) δ H ( r , s )( l , m ) ( β l , m − β n , k )( β l , m − β r , s ) − δ H ( n , k )( l , m ) δ H ( l , m )( l , m ) ( β l , m − β n , k ) (cid:19) , (10) β ( ) l , m = εδ H ( l , m )( l , m ) , (11) β ( ) l , m = ε ∑ n , k δ H ( l , m )( n , k ) δ H ( n , k )( l , m ) ( β l , m − β n , k ) . (12)Eqs. 9-12 are standard equations up to second order in the perturbation parameter ε (higher order contributions,although more complicated in form, can similarly be determined, if needed). The matrix element δ H ( l , m )( n , k ) = < O l , m | δ H | O n , k > = (cid:90) ∞ (cid:90) π O ∗ l , m ( δ H ) O n , k rdrd θ = − k n ∆ (cid:90) ∞ (cid:90) π O ∗ l , m (( ∂ f ( r ) / ∂ r ) cos ( θ ) O n , k r drd θ ; (13)the bra ( < ), ket ( > ) notation signifies a scalar product. This matrix element times ε represents the ellipticity-induced interaction (or coupling) between the fields of the OAM l , m and OAM n , k modes. Inserting Eq. 2 and inte-grating over the azimuthal part, we immediately see that the δ H matrix elements are symmetric ( δ H ( l , m )( n , k ) = δ H ( n , k )( l , m ) ), and nonzero only when | n − l | =
2, a selection rule also pointed out in [7]. Invoking this selectionrule, we obtain simplified forms a ( )( l , m )( l ± , n ) = εδ H ( l ± , n )( l , m ) ( β l , m − β l ± , n ) , (14) a ( )( l , m )( l ± , n ) = ε ∑ k δ H ( l ± , n )( l ± , k ) δ H ( l ± , k )( l , m ) ( β l , m − β l ± , k )( β l , m − β l ± , n ) , (15)and so on. The propagation constant corrections in Eqs. 11 and 12 become β ( ) l , m = , (16) β ( ) l , m = ε (cid:18) ∑ k δ H ( l , m )( l + , k ) δ H ( l + , k )( l , m ) ( β l , m − β l + , k )+ ∑ k (cid:48) δ H ( l , m )( l − , k (cid:48) ) δ H ( l − , k (cid:48) )( l , m ) ( β l , m − β l − , k (cid:48) ) (cid:19) . (17)We note that the mixing coefficient in first order perturbation, Eq. 14, not only depends upon εδ H ( l ± , n )( l , m ) , theellipticity-induced interaction between OAM l , m and OAM l ± , n modes, but also on what we term as a propagator ,which is the inverse of the difference between the squares of the propagation constant of the OAM l , m mode andthe coupled OAM l ± , n mode; larger this difference, smaller this coefficient. The second order coupling of theinput mode OAM l , m to OAM l + , n mode takes place in two steps by first coupling to OAM l + , k , which in turncouples to OAM l + , n ; it is therefore of order ε (all these couplings follow the | ∆ l | = a ( i )( l , m )( l ± , n ) , i ≥
1, will have i δ H matrix elements connected by the ∆ l = ± εδ H (Eq. 6) has the units of the square of the propagation constant,which cancel the units of the propagators, equal in number. Fig. 2 is a pictorial representation of the results inEqs. 14 and 15 for an l = , m = ig. 2: a) An input OAM , mode ( l = , m =
1) couples to the fields of the neighboring
OAM , and OAM , modes in firstorder perturbation; each arrow represents a coupling mediated by the ellipticity-induced interaction, εδ H , times a propagator,which is the inverse of the difference of the propagation constant-squared of the input mode and the connected mode, OAM , or OAM , ; the propagation constant increases from bottom to up, so β , > β , > β , b) Illustration of the second orderperturbation effect, where input OAM , is connected to OAM , through the intermediate OAM , mode as well as the inter-mediate OAM , mode, each sequence of connections making a separate contribution in Eq. 15 through the summation index k . Because two steps (transitions) are involved, the mixing process is of order ε ; note that, in general, there can be more thantwo intermediate states (with the same l = number of of propagators is one less than the number of matrix elements in the numerator, consistent with thedimensionality of the square of a propagation constant.It is also important to note here that the selection rule | ∆ l | = OAM , m mode willcouple to both OAM , n and OAM , n modes ( n ≥ ) in a single step (Eq. 14). But perturbation theory (governed bythe selection rule) also allows the following types of couplings: the input OAM , m mode couples first to to an l = OAM , k , k ≥ l = ( OAM , k (cid:48) , k (cid:48) ≥ ) , which in turn couples (transitions) back tothe OAM , n mode. There are three steps involved, each step contributing a factor of the perturbation parameter, ε ; this mode of coupling is therefore weaker than the single step process (Eq. 14) by ε (recall ε << weaker terms are ignored in arriving at Eq. 14. Weaker terms are also generated in the case of coupling of OAM l , m to OAM l ± in Eq. 15, but they are similarly ignored.In what follows, OAM modes with negative topological charge will be indicated by placing an explicitminus sign. Now, β − l , m = β l , m , a consequence of degeneracy between OAM − l , m and OAM l , m modes; this followsfrom the l dependence of the wave equation [8]. Furthermore, δ H ( − l , m )( − l (cid:48) , n (cid:48) ) = δ H ( l , m )( l (cid:48) , n (cid:48) ) , which is seen fromEq. 13. Thus, when l is replaced by − l in Eqs. 14-17, identical right-hand expressions result. In other words, theperturbation series O (cid:48)− l , m = O − l , m + (cid:48) ∑ n , k a ( )( − l , m )( n , k ) O n , k + (cid:48) ∑ n , k a ( )( − l , m )( n , k ) O n , k + ...., (18)transforms to O (cid:48)− l , m = O − l , m + (cid:48) ∑ n , k a ( )( l , m )( n , k ) O − n , k + (cid:48) ∑ n , k a ( )( l , m )( n , k ) O − n , k + ...., (19)where a ( )( l , m )( n , k ) and a ( )( l , m )( n , k ) are given by Eqs. 14 and 15, as in the + l case. l , m and OAM − l , m Modes
The modes
OAM l , m and OAM − l , m are degenerate in the subspace spanned by these two (orthogonal) modes. How-ever, because of the symmetry of the wave equation with respect to the parameters l and − l , the diagonal elementsof the 2 x ( l , − l ) subspace will be equal to each other, and similarly, the off-diagonal elements.Further, invoking the | ∆ l | = l only; this is due to the fact that the selectionrule requires any transition between them to occur in l steps: l → l − → l − → .... → − l + → − l + → − l ;each step provides a multiplying factor of ε , resulting in a ε l dependence of the off-diagonal element (see Fig. 3as an illustration). In effect, we have a symmetric matrix, which needs to be diagonalized. A formal procedure inAppendix A yields the eigenamplitudes: O ± l , m = / √ ( O l , m ± O − l , m ) , (20)with corresponding eigenvalues ( β (cid:48)± l , m ) = β l , m + β ( ) l , m ± γ l , m . (21) ig. 3: Illustration of the transition from l = , m = l = − , m = | ∆ l | = β ( ) l , m is given by Eq. 17 and γ l , m =( ε ) l ∑ δ H ( l , m )( l − , n ) δ H ( l − , n )( l − , p ) .... ( β l , m − β l − , n )( β l , m − β l − , p ) ........ δ H ( − l + , q )( − l + , j ) δ H ( − l + , j )( − l , m ) .... ( β l , m − β − l + , q )( β l , m − β − l + , j ) ( l > ) ; (22) γ l , m = εδ H ( l , m )( − l , m ) ( l = ) . (23)Each δ H matrix element in the numerator of Eqs. 22 and 23 is a | ∆ l | = l in all; thenumber of propagators is l −
1. The sum runs over all the radial mode solutions of the intermediate OAM states(repeated indices in Eq. 22). Illustrating further, for the l = , m = γ , in Eq. 22 will have products ofmatrix elements δ H ( , )( , n ) , δ H ( , n )( − , p ) , δ H ( − , p )( − , ) and propagators ( β , − β , n ) − , ( β , − β − , p ) − , alongwith the ε multiplying factor (repeated indices imply summation); this is a three-step transition. Note that β − , p = β , p and δ H ( − , p )( − , ) = δ H ( , p )( , ) = δ H ( , )( , p ) due to the symmetries discussed earlier. Physical Interpretation : The linear combinations given in Eq. 20 are the field amplitudes of the corre-sponding
Linearly Polarized (LP) modes, the + sign corresponding to the LP ( a ) l , m spatial mode and the − sign tothe LP ( b ) l , m spatial mode, with intensities proportional to cos ( l θ ) and sin ( l θ ) , respectively; they are, however,now slightly nondegenerate (see Eq. 21). If we further invert Eq. 20, we obtain O ± l , m = / √ ( O + l , m ± O − l , m ) . (24)Consider now an OAM l , m mode, ψ l , m ( z ) = O l , m e i β l , m z (see Eq. 3), incident on the slightly elliptical fiber at z =
0. In view of the result, Eq. 24, this mode will travel down the fiber as an equal superposition of two slightlynondegenerate LP fields with propagation constants, β (cid:48) + l , m and β (cid:48) − l , m , as given by Eq. 21. After traveling a distance L within the fiber, its composition will change to ψ ( e ) l , m ( L ) = √ (cid:16) O + l , m e i β (cid:48) + l , m L + O − l , m e i β (cid:48)− l , m L (cid:17) , (25)where the superscript e refers to the slight ellipticity of the fiber (we have ignored here the interaction with othermodes, which will be done in the next section). We now substitute Eqs. 20 and 21 into the above equation and usethe fact that β (cid:48) + l , m + β (cid:48) − l , m ≈ β l , m , since β ( ) l , m << β l , m . This yields after some algebra ψ ( e ) l , m ( L ) = (cid:16) cos ( π L / L ( π ) l , m ) O l , m + i sin ( π L / L ( π ) l , m ) O − l , m (cid:17) e i β l , m L , (26)where L ( π ) l , m = π ∆ β (cid:48) l , m ≈ πβ l , m γ l , m ; (27) ∆ β (cid:48) l , m = β (cid:48) + l , m − β (cid:48)− l , m ≈ (( β (cid:48) + l , m ) − ( β (cid:48) − l , m ) )) / ( β l , m ) = γ l , m / β l , m (see Eqs. 21). The entering OAM l , m mode oscillatesinto and out of its degenerate partner under the impact of ellipticity; L ( π ) l , m is the 2 π walk-off length, which is thedistance over which a given mode transforms into its degenerate partner, and back into itself. From Eq. 26, wesee that at L = L ( π ) l , m /
4, the input
OAM l , m has partly transformed into its degenerate partner, OAM − l , m , which nowas the same amplitude as the parent mode, OAM l , m . At L = L ( π ) l , m /
2, the input mode has completely transformedinto its degenerate partner (see Fig. 4). This cycle of interconversion of the two degenerate modes continues as L increases further.From Eqs. 27, 22, and 23, we also find that for given topological charge l , L ( π ) l , m ∝ ε − l . Thus, larger theellipticity ε , smaller the 2 π walk-off length. Physically, this implies that as the deviation from fiber circularity(ellipticity ε ) increases, an ellipticity-induced torque acting on the given OAM l , m mode increases, thus shorteningthe distance over which the OAM l , m mode converts into the OAM − l , m mode as depicted in Fig. 4; this conversion,which occurs over half the π walk-off length, involves an OAM transfer of l in magnitude (such a transfer ofOAM also has been discussed in connection with fiber bends [9]). Similarly, for fixed ε , as the required OAMtransfer of l increases with l, we expect the transition to OAM − l , m mode to take place over a longer fiber length. In other words, the 2 π walk-off length L ( π ) l , m increases with l for fixed ε . These features of the 2 π walk-off lengthare further illustrated numerically in Section 6 with reference to a step-index fiber. Fig. 4: Amplitude-squared, equal to cos ( π L / L ( π ) l , m ) and sin ( π L / L ( π ) l , m ) for the OAM l , m and OAM − l , m modes, respectively, asa function of L (see Eq. 26); at L = L ( π ) l , m /
2, the input
OAM l , m mode has transformed completely into its degenerate partner,the OAM − l , m mode. Perturbation fixes the appropriate linear combinations of the degenerate pairs of OAM modes in accordance withEq. 20 (see Appendix A). Therefore, we must work with these linear combinations, and replace the perturbationseries in Eqs. 7 and 19 with the following series: O (cid:48)± l , m = O ± l , m + ∑ i = , k a ( i )( l , m )( n , k ) O ± n , k (28)derived in Appendix A; n = l ± i , i.e., in the above sum, n takes on both the values, l + i and l − i , as in Eqs.14 and 15 due to the | ∆ l | = O ± n , k = / √ ( O n , k ± O − n , k ) are linear combinationsakin to O ± l , m (Eq. 20). Like the latter, they represent the LP fields corresponding to the OAM n , k mode; they arealso slightly nondegenerate due to the slight fiber ellipticity, with ( β (cid:48) + n , k ) − ( β (cid:48) − n , k ) = γ n , k (see Eqs. 22 and 23).Because of the linearity of the wave equation, O (cid:48)± l , m also satisfy the wave equation, Eq. 5, and form a completeset. The O (cid:48) + l , m amplitudes expand in terms of the O + l (cid:48) , m (cid:48) amplitudes, and similarly the O (cid:48) − l , m amplitudes in termsof their corresponding counterparts, O − l (cid:48) , m (cid:48) (Eq. 28). Conversely, the O + l , m amplitudes are expandable in terms of O (cid:48) + l (cid:48) , m (cid:48) amplitudes and similarly O − l , m in terms of O (cid:48) − l (cid:48) , m (cid:48) . Hereafter, the eigenmodes pertaining to amplitudes, O ± l , m will correspondingly be denoted by OAM ± l , m .In the next section, we use the above series to determine the complete expression that includes the mixing ofthe OAM l (cid:48) , m (cid:48) modes, l (cid:48) (cid:54) = − l , not considered in Section 3.1. . OAM Mode-Mixing and Crosstalk For purposes of derivation, consider an input
OAM + l , m mode with amplitude O + l , m entering a straight, slightly ellip-tical fiber. Let ψ +( e ) l , m ( r , θ , z ) | z = L denote the amplitude within the perturbed fiber at a distance L from its entry point( z = r , θ (for brevity reasons), we can then write ψ +( e ) l , m ( L ) = ∑ l (cid:48) , m (cid:48) η +( l , m )( l (cid:48) , m (cid:48) ) O (cid:48) + l (cid:48) , m (cid:48) e i β (cid:48) + l (cid:48) , m (cid:48) L , (29)since the perturbed solutions O (cid:48) + l (cid:48) , m (cid:48) also form a complete set (see Section 3.2). η +( l , m )( l (cid:48) , m (cid:48) ) are the mixing coeffi-cients to be determined. Using the boundary condition, ψ +( e ) l , m ( ) = O + l , m , and assuming orthonormality of O (cid:48) + l , m ’s,we immediately obtain η +( l , m )( l (cid:48) , m (cid:48) ) = (cid:82) ∞ (cid:82) π O ∗ (cid:48) + l (cid:48) , m (cid:48) O + l , m rdrd θ . Inserting Eq. 28, we get η +( l , m )( l (cid:48) , m (cid:48) ) = δ ll (cid:48) δ mm (cid:48) + ∑ i = a ( i )( l (cid:48) , m (cid:48) )( l , m ) δ l (cid:48) , l ± i . (30)In the second term of Eq. 30, l (cid:48) is related to l through the relationship: l (cid:48) = l ± i . Substituting Eqs. 30 and 28 intoEq. 29 yields ψ +( e ) l , m ( L ) = O + l , m e i β (cid:48) + l , m L + ∑ i = , m (cid:48) a ( i )( l , m )( l (cid:48) , m (cid:48) ) O + l (cid:48) , m (cid:48) e i β (cid:48) + l , m L + ∑ i = , m (cid:48) a ( i )( l (cid:48) , m (cid:48) )( l , m ) O + l (cid:48) , m (cid:48) e i β (cid:48) + l (cid:48) , m (cid:48) L + ∑ i = , m (cid:48) ∑ i (cid:48) = , m (cid:48)(cid:48) a ( i )( l (cid:48) , m (cid:48) )( l , m ) a ( i (cid:48) )( l (cid:48) , m (cid:48) )( l (cid:48)(cid:48) , m (cid:48)(cid:48) ) O l (cid:48)(cid:48) , m (cid:48)(cid:48) e i β (cid:48) + l (cid:48) , m (cid:48) L , (31)where l (cid:48) = l ± i and l (cid:48)(cid:48) = l (cid:48) ± i (cid:48) . An identical expression results for an input OAM − l , m mode, except that the − superscript replaces the + superscript everywhere. For an incident OAM l , m mode in which we are interested, wecombine the two results: ψ ( e ) l , m ( L ) = √ (cid:16) ψ +( e ) l , m ( L ) + ψ − ( e ) l , m ( L ) (cid:17) , (32)since O l , m = √ ( O + l , m + O − l , m ) (see Eq. 24). Considering i = ψ +( e ) l , m ( L ) = O + l , m e i β (cid:48) + l , m L + ∑ l (cid:48) = l ± , m (cid:48) a ( )( l , m )( l (cid:48) , m (cid:48) ) O + l (cid:48) , m (cid:48) e i β (cid:48) + l , m L − ∑ l (cid:48) = l ± , m (cid:48) a ( )( l , m )( l (cid:48) , m (cid:48) ) O + l (cid:48) , m (cid:48) e i β (cid:48) + l (cid:48) , m (cid:48) L , (33)where we have used the fact that a ( )( l (cid:48) , m (cid:48) )( l , m ) = − a ( )( l , m )( l (cid:48) , m (cid:48) ) (see Eq. 14); each summation includes only two terms: l (cid:48) = l ±
2. By replacing the + sign of the superscript everywhere in the above equation with the - sign, we obtainthe expression for the incident mode
OAM − l , m : ψ − ( e ) l , m ( L ) = O − l , m e i β (cid:48)− l , m L + ∑ l (cid:48) = l ± , m (cid:48) a ( )( l , m )( l (cid:48) , m (cid:48) ) O − l (cid:48) , m (cid:48) e i β (cid:48)− l , m L − ∑ l (cid:48) = l ± , m (cid:48) a ( )( l , m )( l (cid:48) , m (cid:48) ) O − l (cid:48) , m (cid:48) e i β (cid:48)− l (cid:48) , m (cid:48) L . (34)Combining the two expressions in accordance with Eq. 32, we obtain, up to first order in ε , ψ ( e ) l , m ( L ) = (cid:16) cos ( π L / L ( π ) l , m ) O l , m + i sin ( π L / L ( π ) l , m ) O − l , m (cid:17) e i β l , m L + i ∑ l (cid:48) = l ± , m (cid:48) a ( )( l , m )( l (cid:48) , m (cid:48) ) (cid:16) cos ( π L / L ( π ) l (cid:48) , m (cid:48) ) O l (cid:48) , m (cid:48) + i sin ( π L / L ( π ) l (cid:48) , m (cid:48) ) O − l (cid:48) , m (cid:48) (cid:17) ( sin ( β l , m − β l (cid:48) , m (cid:48) ) L / ) e i ( β l , m + β l (cid:48) , m (cid:48) ) L / . (35) Physical Interpretation : The first two terms correspond to the mixing of the input
OAM l , m mode with itsdegenerate partner, O − l , m mode, which was discussed in detail in Section 3.1. Mixing with the other modes ( l (cid:48) = l ± ) is captured in the second part of the right-hand-side of Eq. 35, where the mixed modes also occurin degenerate pairs and the OAM modes of a degenerate pair oscillate into each other. However, this mixingverall is suppressed by the first order perturbation mixing coefficient. Physically, the incoming OAM l , m modecouples first to the OAM l ± , m (cid:48) mode due to the ∆ l = ± OAM l ± , m (cid:48) mode travels down the slightly elliptical fiber, it also continuously feels the impact of theellipticity-induced torque, changing gradually into its degenerate partner in an oscillatory manner, in accordancewith its characteristic 2 π walk-off length (Eq. 27). A multiplicative sine term is a consequence of the interferenceof the input OAM l , m mode and the coupled OAM l ± , m (cid:48) modes.For the input OAM modes defined by a − l topological charge, we replace Eq. 32 with ψ ( e ) l , m ( L ) = √ ( ψ +( e ) l , m ( L ) − ψ − ( e ) l , m ( L )) , which is consistent with the requirement, ψ ( e ) − l , m ( ) = O − l , m = √ ( O + l , m − O − l , m ) ,the latter following from Eq. 24. Substituting Eqs. 33 and 34, we obtain the same final expression as in Eq. 35,except that l is replaced with − l and l (cid:48) with − l (cid:48) everywhere, and amplitudes, O l , m and O l (cid:48) , m (cid:48) are interchangedwith their degenerate partners, O − l , m and O − l (cid:48) , m (cid:48) , respectively, and l ± − l ± Special Cases
1) For the | l | = l (cid:48) = δ β (cid:48) ± , m = L ( π ) , m (cid:48) = ∞ . Substituting in Eq. 35, we correctly obtain sin ( π L / L ( π ) , m (cid:48) ) = ( π L / L ( π ) , m (cid:48) ) = l = O l , m and the O − l , m modes, respectively, become one and zero, as discussed above. The summation, however,now includes two terms, one corresponding to l (cid:48) = l (cid:48) = −
2, yielding theexpression, ψ ( e ) , m ( L ) = O , m e i β , m L + ia ( )( , m )( , m (cid:48) ) e i π L / L ( π ) , m (cid:48) ( O , m (cid:48) + O − , m (cid:48) ) sin ( β , m − β , m (cid:48) ) e i ( β , m + β , m (cid:48) ) L / e i β , m L . (36)The expression is symmetric in the coupled modes OAM , m (cid:48) and OAM − , m (cid:48) . Physically, the input OAM , m modecouples with the l (cid:48) = ± cosine , sine oscillatory behavior,which cancels out with the net effect of a common (extra) phase factor governed by the 2 π walk-off length. Crosstalk (or charge weight) [3, 6, 9], χ ( l , m )( n , k ) (expressed in dB) for the various component OAM n , k modes of theoutput, ψ ( e ) l , m ( L ) , is given by χ ( l , m )( n , k ) ( L ) = log | < O n , k | ψ ( e ) l , m ( L ) > | = log (cid:12)(cid:12)(cid:12) (cid:90) ∞ (cid:90) π O ∗ n , k ψ ( e ) l , m ( L ) rdrd θ (cid:12)(cid:12)(cid:12) . (37)Upon substitution of Eq. 35, we find the crosstalk with the input mode’s degenerate partner, OAM − l , m , is χ ( l , m )( − l , m ) ( L ) = log sin ( π L / L ( π ) l , m ) . (38)For the content of the original OAM l , m mode remaining within the admixture, Eq. 35, we find χ ( l , m )( l , m ) ( L ) = log cos ( π L / L ( π ) l , m ) . (39)The crosstalk with the neighboring modes, l (cid:48) = l ± ( l (cid:48) > ) , is from Eqs. 35 and 37 χ ( l , m )( l (cid:48) , m (cid:48) ) = log (cid:0) | a ( )( l , m )( l (cid:48) , m (cid:48) ) | cos ( π L / L ( π ) l (cid:48) , m (cid:48) ) sin (( β l , m − β l (cid:48) , m (cid:48) ) L / ) (cid:1) , (40)and with the corresponding degenerate partners, l (cid:48) = − l ∓ ( l (cid:48) < ) , is given by χ ( l , m )( l (cid:48) , m (cid:48) ) = log (cid:0) | a ( )( l , m )( l (cid:48) , m (cid:48) ) | sin ( π L / L ( π ) l (cid:48) , m (cid:48) ) sin (( β l , m − β l (cid:48) , m (cid:48) ) L / ) (cid:1) . (41)For the case of l (cid:48) =
0, which occurs when | l | =
2, we simply set L ( π ) , m (cid:48) = ∞ in Eqs. 40 and 41 to obtain χ ( ± , m )( , m (cid:48) ) = log (cid:0) | a ( )( , m )( , m (cid:48) ) | sin (( β , m − β l , m (cid:48) ) L / ) (cid:1) . The maximum possible value from Eqs. 40 and 41 is χ ( max )( l , m )( l (cid:48) , m (cid:48) ) ≈ log ( | a ( )( l , m )( l (cid:48) , m (cid:48) ) | ) ; (42)his maximum corresponds essentially to setting the magnitude of the sinusoidal factors in Eqs. 40 and 41 tounity; these amplitudes vary rapidly with L due to the small propagation constant differences in the argument ofthe second sinusoidal factor in each of Eqs. 40 and 41. For the l = OAM , m (cid:48) and OAM − , m (cid:48) modes, is given by χ ( , m )( ± , m (cid:48) ) = log ( | a ( )( , m )( , m (cid:48) ) | sin (( β , m − β , m (cid:48) ) L / )) (see Eq. 36).
5. Validity of the Scalar Theory
The crosstalk equations in Section 4.2 along with the expressions for the 2 π walk off length (Eqs. 27, 22, and23) and the admixed neighbor mixing amplitude of Eq. 14 (in first order perturbation), comprise the fundamentalset of equations derived on the basis of scalar perturbation theory. The weakly guiding approximation permittedus to write the vector mode solutions as products of the spatial modes and the polarization states, which thenfacilitated the use of scalar wave equation to study the impact of fiber ellipticity on the spatial modes aloneand their evolution as a function of propagation distance. Inherent in the use of the scalar wave equation is theassumption of the smallness of spin-orbit ( SO ) interaction, which is then neglected.Here we briefly explore the impact of the presence of SO interaction. The SO induced correction to thescalar propagation constant, denoted δ β ( SO )( l , m ) , is of the order of l ∆ / a [8, 16, 17], being of opposite sign for the − l state as compared to the l state. This then implies that the diagonal elements of the 2 x ( l , − l ) subspace (see Section 3.1 and Eq. A. 20), hitherto equal, will tend to become unequal, the change forone diagonal element being of the opposite sign compared to the other. Consequently, this can affect the accuracyof the derived results. In order for our results to be as accurate as possible, we therefore impose the stringentcondition that the splitting of the degenerate spatial modes, 2 γ l , m due to ellipticity ε (Eqs. 21-23) far exceeds thesplitting due to the SO interaction, i.e., the ( l , − l ) degeneracy is still considered broken primarily through theelliptic effects. A simple analysis based on the requirement δ β ( SO )( l , m ) << γ l , m as well as the sufficiency of the firstorder perturbation theory considered in our derivations (see Appendix B, [18]), yields, in an approximate manner,bounds on ε (cid:16) lV k a n (cid:17) / l (cid:16) V (cid:17) << ε << (cid:16) V (cid:17) . (43)The lower bound, denoted ε lo , arises from the requirement of δ β ( SO )( l , m ) << γ l , m , while the upper bound, denoted ε u ,originates in the requirement that | a ( )( l , m )( l (cid:48) , m (cid:48) ) | << ε lo increases with the topological charge l , implying a diminishing region of accuracy, ε u − ε lo ; thisis consistent with increase in SO interaction with l . If V (= kan ( ∆ ) / ) were kept constant (say, by decreasing ∆ and increasing core radius a ), transitioning to a fiber with a larger value of core radius decreases the lower bound, ε lo , thus expanding the region, ε u − ε lo . In general, for large values of V , as in a multimode fiber, the lower bounddecreases but the upper bound becomes tighter. Thus, the accuracy range, as determined by these bounds, dependsupon the fiber parameters and the value of the topological charge l . Conversely, Eq. 43 can be a useful input inthe design of fibers, which suffer slight ellipticity and are governable by scalar perturbation theories due to thegenerally low effects of spin-orbit orbit interaction. Note also that if the above bounds are not strictly adhered to,the derived analytic expressions may suffer in numerical accuracy, but the qualitative behavior (for example, therise in the 2 π walk-off length with topological charge l or its decrement with increasing ellipticity, ε ) will likelyhold; this is also supported by physical reasoning (see Section 3.1). A detailed error analysis is beyond the scopeof the current work.
6. Application to a Step-Index Fiber
As an illustration, we now apply the derived expressions to a step-index fiber, whose solutions are well known andwell studied [8]: O l , m ( r , θ ) = (cid:112) N l , m J l ( p l , m r ) e il θ f or r ≤ a = (cid:112) N l , m J l ( p l , m a ) K l ( q l , m a ) K l ( q l , m r ) e il θ f or r ≥ a , (44)where N l , m , the normalization constant, can be analytically determined [14]; J l and K l are the Bessel and themodified Bessel functions, respectively; p l , m = (cid:113) k n − β l , m and q l , m = (cid:113) β l , m − k n . The index profile f ( r ) is step-function equal to zero for r ≤ a and equal to 1 for r > a . As a result, ∂ f ( r ) / ∂ r = δ ( r − a ) . The matrixelement for the δ H operator defined in Eq. 13 is now given by δ H ( l , m )( l ± , m (cid:48) ) = − ( k n ∆ ) J l ( p l , m a ) J l ± ( p l ± , m (cid:48) a ) (cid:113) N (cid:48) l , m (cid:113) N (cid:48) l ± , m (cid:48) , (45)where N (cid:48) n , k = N n , k / π a is dimensionless. Propagation constants required in the calculations are determined usingthe well-known characteristic equation for the scalar modes of a step-index fiber [8,19]. All numerical calculationsare done in MatLab. We assume a = µ m , n = . , n = . ∆ = ( n − n ) / ( n ) = . <<
1. For a wavelength, λ = . µ m , normalized frequency V = π a ( n − n ) / / λ = . OAM , , OAM , , OAM − , , OAM , , OAM − , and OAM , modes are supported. From Eq. 23 and Eq. 22, γ , = εδ H ( , )( , − ) (46)and γ , = ε (cid:16) ( δ H ( , )( , ) ) β , − β , + ( δ H ( , )( , ) ) β , − β , (cid:17) . (47)Now using Eqs. 27, 22, and 23, we can calculate the 2 π walk-off lengths, which are displayed in Table 1 for variousvalues of ellipticity. Larger the ellipticity ε , smaller the 2 π walk-off length L ( π ) l , m due to the ε − l dependence. Table 1: The 2 π walk-off length, L ( π ) l , m , specified in meters, as a function of ellipticity ε for different input modes, OAM l , m ; itvaries as ε − l for fixed l , m (see Eqs. 27, 22, and 23); normalized frequency V = . l , m ε = . ε = . ε = . ε = . ε = . l , for the same ellipticity ε value, larger the 2 π walk-off length(Section 3.1). This is what we observe in Table 1. Note also that the ellipticity values for the l = ε lo = . ε u = . l = ε lo = . Input:
OAM , The
OAM , input mode can only mix with OAM − , due to the ∆ l = ± π walk-off length). Referring to Eq. 35, excluding l (cid:48) = −
1, the only other allowedvalue of l (cid:48) is 3, which is not supported by the fiber. So there is no other mode the input mode OAM , can mix with.Input: OAM , Besides undergoing the transformation into the degenerate partner, the
OAM − , mode, the OAM , modealso mixes with the modes OAM , and OAM , with the amplitudes given by (see Eq. 35)1) | < O , | ψ ( e ) , > | = | a ( )( , )( , ) sin (( β , − β , ) L / ) | = . ε | sin ( L ) | | < O , | ψ ( e ) , > | = | a ( )( , )( , ) sin (( β , − β , ) L / ) | = . ε | sin ( L ) | The numerical coefficient of the amplitude is larger in 2) than in 1) due to the closer proximity of the pairof modes
OAM , , OAM , in their propagation constant values as compared to the mode pair: OAM , , OAM , ;these propagation constant differences are reflected in the numerical part of the argument of the sine functions.The amplitudes vary rapidly with L , but the maximum amplitude corresponds to setting the sine term above tounity. These expressions are used in the crosstalk calculation in Section 6.1.2. able 2: Crosstalk, χ ( , )( l (cid:48) , m (cid:48) ) (in dB) for the various component OAM l (cid:48) , m (cid:48) modes within the OAM , output mode mixture, ψ ( e ) , as a function of L (see Section 4.2); the ellipticity ε is fixed at 0.025. The 2 π walk-off length L ( π ) , = . m (see Table 1). l (cid:48) , m (cid:48) L=0.10m L=0.25m L=0.50m L=1.00m L=1.50m2,1 -0.32 -2.12 -12.91 -0.94 -3.98-2,1 -11.53 -4.12 -0.23 -7.12 -2.220,1 -44.33 -44.89 -47.33 -44.22 -46.950,2 -30.48 -27.30 -22.87 -25.07 -30.86
For ε = . L = . m , we see thatthe output mixture is practically all OAM , mode with little OAM − , mode content. However, at L = . m ,roughly one quarter of the 2 π walk-off length value, the contents of the OAM , and its degenerate partner, OAM − , mode are approximately the same, with the latter becoming the major constituent of the output mix-ture at L = . m (roughly half the 2 π walk-off length). At L = m , slightly less than the 2 π walk-off length, the OAM , mode is the main contributor to the mixture, and this cycle of mode conversion from the OAM , modeinto OAM − , mode, and back into the original OAM , mode, continues as L is increased beyond the 2 π walk-offlength. The contents of the other admixed modes l (cid:48) (cid:54) = ± l remain very low being (upper) bounded by values of − . dB and − . dB (calculated from Eq. 42) for χ ( max )( , )( , ) and χ ( max )( , )( , ) , respectively. If we now adopt a crosstalk criterion of χ ( , )( l (cid:48) , m (cid:48) ) ≤ − dB for successful OAM , transmission, then the maxi-mum propagation distance, denoted L ( max ) , , that is allowed is given by setting 10 log ( sin ( π L ( max ) , / L ( π ) , )) = − − l degenerate OAM component (see Eq. 38). This gives rise tomultiple solutions due to the multi-valued sin − function: L ( max ) l , = . m , L ( max ) l , = ( ± . + n ) L ( π ) l , ( m ) ,where n > L must satisfy the constraint L ≤ L ( max ) , = . m , ( nL ( π ) , − . m ) < L ≤ ( nL ( π ) , + . m ) , where n ( > ) is an integer. Because L ( π ) , = . m , this im-plies permitted transmissions at most of the order of a few meters suitable for inter-shelf/rack distances within datacenters. Furthermore, the 2 π walk-off length of the degenerate partner is identical and the nondegenerate modes, OAM , and OAM , , which can only mix with the OAM , and the OAM − , modes due to the ∆ l = ± − dB (see Table 2). Consequently, under this criterion we can havesimultaneous transmission of the four spatial modes, OAM , , OAM − , , OAM , , and OAM , in a multiplexed en-vironment, thereby increasing the data transmission capacity four-fold. Additional tables can be constructed fordifferent sets of ellipticity and fiber parameters to gain insight into the constraints imposed by ellipticity in thedesign of such fibers. We assume a = µ m , n = . n = . ThorLabs multimodefiber. ∆ = ( n − n ) / ( n ) = . λ = . µ m . The normalized frequency V is 22.3. Although themaximum allowed value of l is 18, we consider l up to 6. Fig. 5a shows the effective refractive indices for thevarious OAM modes. The values decrease with increasing l and m . For fixed l , the spacing between consecutive m indices also increases, the increase being larger for larger l values. The effective refractive index difference, ∆ n e f f , is the smallest for the OAM , and the OAM , mode pair and equals 0 . x − . However, the ∆ l = ± OAM , and the OAM , modes, which are very close in proximity ( ∆ n e f f = . x − ) but also satisfy the ∆ l = ± a ( )( , )( , ) , is likely to be larger as compared with the mixing coefficientof the modes OAM , and OAM , that also satisfy the ∆ l = ± ∆ n e f f = . x − . In general, the proximity of the OAM l , and the OAM l − , modes in their n e f f values (seeFig. 5a) causes the latter to be the predominantly admixed neighboring mode.Fig. 5b shows L ( π )( l , ) as a function of ellipticity ε for different values of topological charge l . It is calcu-lated using Eqs. 27, 22, and 23. It decreases as ellipticity ε is decreased, the drop being sharper for higher l values; this is due to the ε − l dependence. We note here that in the limit ε →
0, the 2 π walk-off length approachesinfinity, consistent with the absence of crosstalk (see Eqs. 38, 41 and 14) for a perfectly round fiber ( ε = ) . ig. 5: a) The effective refractive indices, n e f f ’s shown with a cut-off of 1.456, for most modes up to l =
6; the values decreasewith increasing l and m , which is shown explicitly in parentheses for the l = π walk-off length, L ( π )( l , ) , as afunction of ellipticity ε for different values of l ; for fixed l , L ( π )( l , ) varies inversely as ε l c) The 2 π walk-off length, L ( π )( l , ) , as afunction of topological charge l for different values of ε d) χ ( max )( l , )( l − , ) , the maximum possible crosstalk of OAM l , mode withthe neighboring mode, OAM l − , , as a function of ε for different l values; see text. n Fig. 5c, the ellipticity ε is fixed. The 2 π walk-off length rises with l , the rise being sharper with l , whichimplies that the higher l value modes will have relatively less crosstalk with their degenerate partners (see Eq. 38). Table 3: The lower bound, ε lo and the upped bound ε u for different values of l ; ε u is independent of l (see Eq. 43). l , m ε lo ε u . X − a of the core ofthe multimode fiber compared to that of the few mode fiber causes the SO interaction to reduce; on the otherhand, the elliptic effects leading to splitting of the degenerate modes are enhanced due to the relatively smallerpropagation constant differences on account of a larger value of V . The preferred region of ε (Eq. 43) expands toaccommodate lower values of ε , but diminishes as l increases; see Table 3, where the lower bound ε lo increaseswith l , consistent with a rising value of the ”undesired” SO interaction (of order l ∆ / a ) ; the desired ellipticityrange, ε u − ε lo , therefore, reduces with l . If we were to increase the core radius to 62 . µ m correspondingto another commercially available fiber, according to Eq. 43 and the fact that V = kan ( ∆ ) / , the lowerbound ε lo would decrease, for fixed l , by a factor ( / . ) / l ( / . ) ; the reduced ε lo values then would be0.074, 0.014, 0.020 for l = , ,
5, respectively. Thus, by increasing the core radius, we can accommodate lowervalues of ε at higher values of l . Such analyses can be useful in the design of fibers with low spin-orbit interaction.In Table 4, we show the crosstalk experienced by the input OAM , and OAM , modes in a fiber with el-lipticity ε = . , (which lies within the bounds defined by Table 3). While these modes change into theirrespective degenerate partners, they also mix with the neighboring modes, OAM l ± , m (cid:48) with amplitudes propor-tional to a ( )( l , m )( l ± , m (cid:48) ) determined from Eq. 14. The crosstalk values displayed in Table 4 are all seen to be belowthe maximum possible values of χ ( max )( , )( , ) = − . dB and χ ( max )( , )( , ) = − . dB (Eq. 42). The 2 π walk-offlength for the l = . m . We note from the table that at L = . m , the OAM , mode is essentiallypure with very little OAM − , content. However, at L = . m , which is approximately one-quarter the 2 π walk-offlength, the contents of the OAM , mode and its degenerate partner are roughly equal. At L = . m (approximatelyhalf the 2 π walk-off length), the OAM , is basically converted fully into the OAM − , mode. Similarly, at L = m (slightly less than the 2 π walk-off length), most of the OAM − , mode has changed back into the OAM , mode.This oscillatory behavior continues, with a period equal to 4 . m , the 2 π walk-off length. For the l = π walk-off length is 0 . m ; the changes here are more rapid on account of the smaller value of the 2 π walk-offlength. Table 4: Crosstalk, χ ( l , m )( l (cid:48) , m (cid:48) ) (in dB) (see Section 4.2); m = l = ε is fixed at 0 . L is the length of thefiber traversed by the OAM l , mode input at one end of the fiber; l (cid:48) = l − l (cid:48) , m (cid:48) L=0.5m L=1.0m L=2.0m L=4.0m L=5.0m3,1 -0.64 -2.77 -24.92 -0.06 -1.94-3,1 -8.65 -3.26 -0.01 -18.91 -4.441,2 -20.59 -23.07 -19.11 -34.04 -18.65-1,2 -16.18 -17.55 -15.75 -25.99 -17.282,1 -1.97 -11.35 -1.38 -6.82 -0.17-2,1 -4.38 -0.33 -5.66 -1.01 -14.060,2 -17.86 -16.69 -19.92 -16.26 -30.70
If we now consider a fiber which is perfect, except for ellipticity, and adopt a somewhat relaxed criterionof crosstalk < − dB with any admixed mode, then this criterion translates into a requirement of a maxi-mum permitted propagation distance, L ( max ) l , , given by 10 log sin ( π L ( max ) l , / L ( π ) l , ) = −
14; this implies multi-le solutions (as in Section 6.1.3) and leads to the constraint that the propagation distance L must satisfy: L ≤ . L ( π ) l , , ( n − . ) L ( π ) l , < L < ( n + . ) L ( π ) l , , l = ,
3. For example, L=4.40m satisfies the inequal-ity for both l = l = ( n = , l = ,
2, respectively). This has the implication that we can multiplex
OAM , , OAM − , , OAM , , OAM − , and transmit without crosstalk exceeding − dB when L = . m ; this cor-responds to an inter-shelf/rack distance in a data center.
7. Further Remarks
1) Because the weakly guiding approximation causes the vector modes to be expressed as products of the spatialOAM mode considered here and the circular polarization (right or left) (Section 2), the impact of fiber ellipticityis then the product of the impact of ellipticity on the spatial mode and the polarization mode, evaluated separately.In general, the initial circular polarization would become a linear combination of the left and right circularpolarizations; the overall final state (including polarizations) would be the product of the final spatial mode stateas determined by our analysis and the final polarization state not determined here; this would, for example, resultin a linear combination of the quartets: O l , m ε + , O − l , m ε − , O l , m ε − , O − l , m ε + , translating, respectively to the vectormodes: HE l + , m , HE − l − , m , EH l − , m , EH − l + , m ( | l | > ) . Our intent here is to treat the impact of ellipticity on thespatial mode only, meaning we are not concerned with the final polarization state of the output modes. In theexperimental domain, this corresponds to separating out the spatial modes, as is done, for example, in [24, 25]using a mode sorter, and measuring their individual intensities, without any regard to their polarization state.2) The solutions, O (cid:48) ± l , m , Eq. 28, are the eigenmodes of the scalar wave equation: ( H + εδ H ) O (cid:48) ± l , m = ( β (cid:48) ± l , m ) O (cid:48) ± l , m .Since these field amplitudes are dominated by the O ± l , m amplitudes, these eigenmodes would display the fields ofthe LP modes (Section 3.1). Indeed such LP mode-like fields have been observed in numerical simulations usingthe finite -element solver COMSOL for the slightly elliptical ring-core fibers in [5]. This qualitative agreementgives further credence to our analytic results based on perturbation theory, and suggests additionally that ourresults, Eqs. 20 and 35, are of a general nature applicable to other fibers as well. Furthermore, the rapid rise in thevalue of the 2 π walk-off length, predicted by our analytic expressions, as the topological charge l of the incomingmode is increased (keeping ellipticity ε constant) or the ellipticity ε is decreased (keeping l constant), is also seenin the finite-element calculations in [3, 6].3) Because the output intensity pattern of the final state is dominated by the presence of the OAM l , m and OAM − l , m modes in the output mixture, the output intensity pattern is essentially given by I ( r , θ ) = | ( cos ( π L / L ( π ) l , m ) O l , m + i sin ( π L / L ( π ) l , m ) O − l , m ) | (see Eq. 35), which, upon insertion of O l , m ( r , θ ) = ( / √ N l , m ) F l , m ( r ) e il θ and O − l , m ( r , θ ) = ( / √ N l , m ) F l , m ( r ) e − il θ , reduces to ( + sin ( π L / L ( π ) l , m ) sin ( l θ )) F l , m ( r ) / N l , m . This result represents tilted l -lobed LP mode patterns ( [9, 21, 22]). For example, at L = L ( π ) l , m /
4, intensity I ∝ cos ( l ( θ − π / l )) ,implying tilt angles of π / ( l ) . For l =
1, the tilt angle is π /
4, a result also predicted in [2].
8. Summary and Discussion
Working in the weakly guiding approximation, in which the vector modes reduce to products of the spatial OAMmodes and the associated polarizations, we have focused on the spatial modes and used the scalar wave equationto develop perturbation theory for spatial OAM mode mixing and the ensuing OAM modal crosstalk due to slightellipticity in a fiber. The developed perturbation technique provides insight into the mechanism for mode mixing.A fundamental mixing selection rule ( ∆ l = ±
2) makes the problem tractable and leads to the derivation of analyticexpressions for mode-mixing and consequently the crosstalk. The expressions include an explicit formula forthe 2 π walk-off length, which is essential in determining the degenerate OAM mode pair mixing. The derivedexpressions embody complete mathematical dependence on topological charge, ellipticity, and the length of thetraversed fiber, enabling detailed quantitative analyses of the ellipticity-induced crosstalk, and thereby facilitatingthe study of optimal fiber parameters for minimal crosstalk.Scalar perturbation theory is a well-established theory to study perturbations in a given system [13]. Thistheory, as developed and presented here, provides a convenient and useful framework to explore the impact ofellipticity. A generic but vital feature in the use of this theory is the negligibility of the polarization effects,especially the spin-orbit interaction that gives rise to differences in the propagation constants of the vector modes, HE l + , m and EH l − , m . In this work, we subsequently explore the effect of spin-orbit interaction to estimatebounds on the values of ellipticity for which the derived results are most valid (accurate). The validity rangeexpands with increasing values of normalized frequency, V , as in a multimode fiber (larger core radius implieselatively smaller spin-orbit interaction), and dwindles as the topological charge is increased, consistent with thefact that the spin-orbit interaction becomes significant at higher values of l and thus cannot be ignored. As a result,the numerical results based on the developed theory are most accurate for low values of topological charge (asdetermined by the fiber parameters). We have illustrated the utility of the analytic expressions and their validitywith application to a step-index few mode fiber and a multimode fiber.Prior work has focused on the use of computationally intensive finite-element methods that would nor-mally require high levels of precision to obtain detailed relationships with fiber parameters. Therefore, theavailability of analytic expressions based on the presented scalar theory, along with their utility constraints andthe provided insight, serve as a valuable tool in the analysis and design of fibers for OAM mode propagation.The technique, as applied to slightly elliptical fibers, is novel and detailed, and, to our knowledge, has not beenpresented before. This technique also serves as a precursor to techniques incorporating groups of degeneratemodes as in graded-index fibers. Appendix
A. Derivation of the Mixing of the
OAM l , m and OAM − l , m Modes due to Fiber Ellipticity
The derivation here follows closely that for the fiber bend in [9]. The states
OAM l , m and OAM − l , m are degenerate.Therefore, there are an infinite number of combinations of these states that satisfy the unperturbed scalar waveequation (Eq. 1). However, there is a unique pair of these linear combinations, which are orthonormal, and ap-propriate to the perturbation under consideration. We now seek these. Due to the Hermitian nature of H and δ H ,these linear combinations are related to the above two degenerate states via a unitary transformation in the ( l , − l ) subspace. Consequently, we write O + l , m = U O l , m + U O − l , m , (A.1) O − l , m = U O l , m + U O − l , m , (A.2)where U i , j are the elements of the 2 x U . We now replace O l , m in Eq. 7 with O + l , m , and O − l , m inEq. 19 with O − l , m , and rewrite the two perturbation series in a standard procedure [13–15] as O (cid:48)± l , m = O ± l , m + ∑ i = ∑ n (cid:54) = ± l , k a ± ( i )( l , m )( n , k ) O n , k , (A.3)where index i signifies the perturbation order. The two individual series are labeled by + and − signs, with n (cid:54) = + l applying to the former and n (cid:54) = − l applying to the latter. The coefficients, a ± ( i )( l , m )( n , k ) , are defined in the sameway as a ( i )( l , m )( n , k ) (see Eqs. 14 and 15), with the difference, however, that the matrix elements, δ H ( n , k )( ± l , m ) = < O n , k | δ H | O ± l , m > (see Eq. 13) are replaced with δ H ± ( n , k )( l , m ) = < O n , k | δ H | O ± l , m > . Substituting Eqs. A.1 and A.2,we obtain δ H +( n , k )( l , m ) = < O n , k | δ H | O + l , m > = U δ H ( n , k )( l , m ) δ n , l ± + U δ H ( n , k )( − l , m ) δ n , − l ± (A.4)and δ H − ( n , k )( l , m ) = < O n , k | δ H | O − l , m > = U δ H ( n , k )( l , m ) δ n , l ± + U δ H ( n , k )( − l , m ) δ n , − l ± . (A.5)Replacing δ H ( n , k )( l , m ) with Eqs. A.4 and A.5 in the perturbation series, Eq. A.3, transforms it to O (cid:48)± l , m = O ± l , m + ∑ i = ∑ k a ( i )( l , m )( n , k ) O ± n , k (A.6)where O + n , k = U O n , k + U O − n , k , (A.7) O − n , k = U O n , k + U O − n , k , (A.8)and n = l ± i (both these values are included in the sum in Eq. A.6). Starting from Eq. A.3, we have thus obtaineda new form of the series, expressed entirely in terms of linear combinations of the degenerate states. Furthermore,these linear combinations, regardless of the topological charge, are described uniquely by the elements of a singleunitary matrix U (to be determined later). The mixing coefficients of O ± n , k are the same as those in the originalperturbation series for O (cid:48)± l , m .Consider now ( H + εδ H ) O + (cid:48) l , m = ( β (cid:48) + l , m ) O + (cid:48) l , m . (A.9)riting ( β (cid:48) + l , m ) = β l , m + δ β (cid:48) l , m , where δ β (cid:48) l , m is to be determined along with matrix U , and substituting this expres-sion in Eq. A. 9, we obtain ( H − β l , m ) O + (cid:48) l , m = ( δ β (cid:48) l , m − εδ H ) O + (cid:48) l , m . (A.10)Taking the scalar product on the left with O l (cid:48) , m ( l (cid:48) = ± l ) , we have < O l (cid:48) , m | H − β l , m | O + (cid:48) l , m > = < O l (cid:48) , m | δ β (cid:48) l , m − εδ H | O + (cid:48) l , m > . (A.11)Now < O l (cid:48) , m | H − β l , m | O + (cid:48) l , m > = < ( H − β l , m ) O l (cid:48) , m | O + (cid:48) l , m > due to the Hermiticity of H − β l , m . But ( H − β l , m ) O l (cid:48) , m =
0. Therefore, the LHS of Eq. A.11 equals zero. It follows then < O l (cid:48) , m | δ β (cid:48) l , m − εδ H | O + (cid:48) l , m > = . (A.12)Inserting Eqs. A. 6, A.1, and A.7 into Eq. A.12, we obtain < O l (cid:48) , m | εδ H − δ β (cid:48) l , m | ( U O l , m + U O − l , m ) > + ∞ ∑ i = ∑ k a ( i )( l , m )( l ± i , k ) < O l (cid:48) , m | εδ H − δ β (cid:48) l , m | U O l ± i , k + U O − ( l ± i ) , k > = . (A.13)Now l (cid:48) = ± l leads to two linear equations in U and U :1) l (cid:48) = l ε ∞ ∑ i = ∑ k a ( i )( l , m )( l ± i , k ) ( U δ H ( l , m )( l ± i , k ) + U δ H ( l , m )( − ( l ± i ) , k ) ) − U δ β (cid:48) l , m = . (A.14)2) l (cid:48) = − l ε ∞ ∑ i = ∑ k a ( i )( − l , m )( l ± i , k ) ( U δ H ( − l , m )( l ± i , k ) + U δ H ( − l , m )( − ( l ± i ) , k ) ) − U δ β (cid:48) l , m = . (A.15)The above homogeneous equations have a non-trivial solutions if and only if the determinant of the coefficients of U and U is zero. Invoking the real and symmetric nature of δ H matrix elements and the fact that the mixingcoefficients for l and − l cases are identical in values (see Section 3), it is easy to see that the corresponding matrixis of the form, M − δ β (cid:48) l , m I , where matrix M is symmetric (with diagonal elements equal) and I is the 2 x OAM l , m and OAM − l , m modes, the off diagonal elements of the M matrix must be non-zero. Examining the coefficient of U in Eq. A.14, which is an off diagonal element of the2 x δ H ( l , m )( − ( l ± i ) , k ) = δ H ( − l , m )( l ± i , k ) = δ H ( l ± i , k )( − l , m )) , using the symmetric propertiesof the δ H matrix elements (see Section 3). Invoking the | ∆ l | = i = l − l ± i ; the + sign gives an infeasible solution). The value of thisoff diagonal term, denoted γ l , m , is given by γ l , m = ∑ k ε a ( l − )( l , m )( − l + , k ) δ H ( − l + , k )( − l , m ) . (A.16)Expanding out the a ( l − )( l , m )( − l + , k ) coefficient as a product of l − δ H elements and the l − γ l , m =( ε ) l ∑ δ H ( l , m )( l − , n ) δ H ( l − , n )( l − , p ) .... ( β l , m − β l − , n )( β l , m − β l − , p ) ........ δ H ( − l + , q )( − l + , k ) δ H ( − l + , k )( − l , m ) .... ( β l , m − β − l + , q )( β l , m − β − l + , k ) ( l > ) . (A.17)Identical result follows if we examine the other off diagonal term of the M matrix (coefficient of U in Eq. A.15).For l = γ l , m = εδ H ( l , m )( − l , m ) ( l = ) . Furthermore, we observe from Eq. A.14 that in the lowest order, whichcorresponds to i =
1, the coefficient of U , denoted κ l , m , is given by κ l , m = ∑ k ε a ( )( l , m )( l + , k ) δ H ( l , m )( l + , k ) + ∑ k (cid:48) a ( )( l , m )( l − , k (cid:48) ) δ H ( l , m )( l − , k (cid:48) ) . (A.18)ubstituting Eq. 14, we find κ l , m = ε (cid:18) ∑ k δ H ( l , m )( l + , k ) δ H ( l + , k )( l , m ) ( β l , m − β l + , k )+ ∑ k (cid:48) δ H ( l , m )( l − , k (cid:48) ) δ H ( l − , k (cid:48) )( l , m ) ( β l , m − β l − , k (cid:48) ) (cid:19) , (A.19)which is identified with β ( ) l , m (see Eq. 17); this is not surprising, since κ l , m is the diagonal term of the 2 x M in the ( l , − l ) subspace. It is the nonzero nature of the off diagonal term, γ l , m , which breaks the degeneracy.Matrix M can now be written as M = (cid:34) β ( ) l , m γ l , m γ l , m β ( ) l , m (cid:35) . (A.20)The eigenvalue equation is ( M − δ β (cid:48) l , m I ) φ =
0, where δ β (cid:48) l , m is the eigenvalue of matrix M , and φ (a 2 x M − δ β (cid:48) l , m I ) to zero then yields the twoeigenvalues: δ β (cid:48)± l , m = β ( ) l , m ± γ l , m . (A.21)That is, ( β (cid:48) ± l , m ) = β l , m + β ( ) l , m ± γ l , m . (A.22)The corresponding eigenvectors are √ (cid:20) (cid:21) and √ (cid:20) − (cid:21) , which are identified with the two column vec-tors of the U matrix. That is, U = √ , U = √ , U = √ , and U = − √ . The unitary matrix U isindependent of l . Inserting the numerical values of the appropriate U i j elements in Eqs. A.1 and A.2, weobtain O + l , m = √ ( O l , m + O − l , m ) and O − l , m = √ ( O l , m − O − l , m ) . Similar insertions in Eqs. A.7 and A.8 yield O + n , k = √ ( O n , k + O − n , k ) and O − n , k = √ ( O n , k − O − n , k ) , where n = l + i , i >
0. As a result, we now know theentire perturbation series, Eq. A.6.Below, we indicate a few example forms for γ l , m , using Eq. A.17. For l = γ , m = ε ∑ i ( δ H ( , m )( , i ) )( δ H ( , i )( − , m ) ) β , m − β , i . (A.23)For l = γ , m = ε ∑ i , j ( δ H ( , m )( , i ) )( δ H ( , i )( − , j ) )( δ H ( − , j )( − , m ) )( β , m − β , i )( β , m − β , j ) , (A.24)where we have used the fact that β − , j = β , j .We can similarly write down expressions for higher values of l . B. Derivation of the Bounds on Ellipticity
The spin-orbit ( SO ) interaction correction to the scalar propagation constant, denoted δ β ( SO )( l , m ) , is of the orderof l ∆ / a [8, 16, 17]. These corrections are of opposite sign for the spin-orbit aligned case (e.g., + l , S = + − l , S = + S denotes the spin of the OAM mode, with S = ± x γ l , m due to ellipticity ε (Eq. 22), i.e., the ( l , − l ) degeneracy is brokenprimarily through the ellipticity effects. By examining the analytic form for γ l , m ( Eq. 22), we estimate thecondition under which this may hold.From Eq. 13, we see that εδ H is of the order of ε ∆ k n . The magnitude of γ l , m in Eq. 22 is deter-mined by l such factors in the numerator and l − β l (cid:48) , m (cid:48) − β l (cid:48) ± , m (cid:48)(cid:48) ≈ kn ( β l (cid:48) , m (cid:48) − β l (cid:48) ± , m (cid:48)(cid:48) ) . We now recall that the total number of ( l , m ) pairs in a fiber (notcounting polarization states and the degenerate counterparts) is approximately α V / V ) [8, 19],here α varies from 0.5 for a parabolic fiber to 1 for a step-index fiber. This implies an average modespacing of k ( n − n ) / ( α V / ) that can for practical purposes for few mode (small V ) as well as multimode(large V ) fibers be represented approximately by k ( n − n ) / V ; V is the normalized frequency equal to ka ( n − n ) / = kan ( ∆ ) / . Thus, γ l , m ∼ ( ε ∆ k n ) l / ( k n ( n − n ) / V ) l − = ∆ k n ε l ( V / ) l − . Imposing thecondition δ β ( SO )( l , m ) << γ l , m now leads to the inequality (cid:16) lk a n ( V / ) l − (cid:17) << ε l . (B.1)We further derive an upper bound from the requirement that the magnitude of the first order amplitudes a ( )( l , m )( l (cid:48) , m (cid:48) ) (Eq. 14) be much smaller than unity [13] for results to be valid in first order perturbation. From the above dis-cussion, this implies ε ∆ k n << k n ( n − n ) / V , yielding ε << / V . (B.2)Combining Eqs. B.1 and B.2, we obtain (cid:16) lV k a n (cid:17) / l (cid:16) V (cid:17) << ε << V . (B.3)We note here that the left hand side inequality becomes less stringent (compared to Eq. B.1) for higher values of l . eferences
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