Light Spins of Cylindrical Electromagnetic Waves and their Jumps across Material Interfaces in the Presence of Energy Exchange

We investigate light spins for cylindrical electromagnetic waves on resonance. To this goal, we consider both a dielectric cylinder of infinite length immersed in vacuum and a cylindrical hole punched through a dense dielectric medium. In order for waves of constant frequencies to be established through lossless media, energy absorption is allowed in the surrounding medium to compensate for radiation loss. The dispersion relation is then numerically solved for an asymmetry parameter implying a balance in energy exchange. Numerical studies are performed by varying parameters of refractive index contrast, azimuthal mode index, and size parameter of a cylindrical object. The resulting data is presented mostly in terms of a specific spin, defined as light spin per energy density. This specific spin is found to be bounded in its magnitude, with its maximum associated with either optical vortices or large rotations. Depending on parametric combinations, the specific spin could not only undergo finite jumps across the material interface but also exhibit limit behaviors.


Introduction
Among various measures characterizing electromagnetic (EM) waves, energy density is the most fundamental one, but it is scalar.Several additional measures have so far been utilized in order to illustrate the vector nature of Maxwell's equations.They are Poynting vector, angular momentum, and chirality (or helicity) for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14].Polarizations and polarization ellipses are in between scalars and vectors, depending on their definitions [3,10,14].The angular momentum (AM) of EM waves can be separated into its orbital and spin parts [12,15,16].The orbital AM is extrinsic, since it depends on the distance vector between a coordinate center and the point of application.In contrast, the spin AM is intrinsic, since it is not required to specify such a distance vector [3].
The spin AM of EM waves is henceforth called the "light spin" in this study in order to be differentiated from the electron spin.Light spin is similar to electron spin in the sense that both are intrinsic from a macroscopic viewpoint.However, the light spin is different from the electron spin, since light spin takes on continuous values, whereas the electron spin takes essentially two discrete values (either an up-spin or a down-spin).For our cylindrical EM waves, light spin turns out to be proportional to the light-induced fictitious magnetic field, which has been recognized in the area of light trapping [8].
Traditionally, light spin has been suitably normalized by taking the energy density as a reference [3,10].This light spin per energy density will be henceforth called the "specific spin" for short [9,17].This specific spin turns out be bounded in its magnitude by unity so that it can be compared to the electron spin [13].As an example for revealing typical features of a specific spin, we take cylindrical EM waves rotating around a cylindrical object [1].For instance, optical fibers and nano-scale probes rely on wave propagations along the cylindrical axis [10,18].Even when waves are allowed to propagate only on the cross-sectional plane of a wire, there are numerous technological applications, for instance, involving whispering-gallery modes [19,20].
Most of these applications have been successfully analyzed as regards the energy transfer from a cylindrical object into its surrounding environment via the century-old radiation condition [8,21].However, a rotational wave under our study cannot be maintained at constant frequency, because of the unending one-way radiation loss into the surrounding.In such a case, frequency is considered as a complex variable in order to account for wave attenuations even if all the participating media are lossless.
Difficulties arise for complex frequency, since many defining notions for EM waves get blurred.For instance, the cherished formula for time-averaged Poynting vector becomes useless.Even the formula for the energy density of EM waves gets fuzzy for lossy dielectric media and even more difficult for lossy metals.
Our remedy for this difficulty is to admit both energy radiation and absorption between a cylindrical object and its surrounding medium [17].In the context of exterior boundary value problems [21], not only outgoing waves but also incoming waves are assumed to exist [22].With such two-way energy exchanges, we have recently carried out an investigation into plasmonic resonances around a lossy metallic nanowire [17].As a result, we were able not only to obtain a useful explanation of wave dynamics but also to prove the aforementioned boundedness of specific spin.
In fact, this coexistence of energy radiation and absorption is not an entirely new concept though.As an example, optical trapping of ions relies on energy supply from outside through laser illumination [8,23].As another example, optical gain media for metamaterials act as energy sources, thus compensating energy dissipation by metallic constituents [6,7,24].In a similar fashion, sunlight is absorbed from the environment onto solar cells, for which design scientists try to minimize the inevitable energy radiation associated with reflection and scattering [7,9,11,20].
Our focus here lies in presenting unusual behaviors of the specific spin for typical parameter sets.To this goal, two configurations are considered.Firstly, we consider a solid cylinder with its interior possessing a larger dielectric constant than that of its exterior.For instance, a silica cylinder is employed for light trapping [8].Secondly, as a conjugate configuration to the solid cylinder, we consider a cylindrical hole with its interior having a smaller dielectric constant than that of its exterior.In this study, both dielectric media are assumed lossless for the sake of simplicity of analysis.In addition, the effects of the cylinder's size with respect to the wavelength will be investigated.By this way, the jumps in the specific spin across the cylindrical dielectric-dielectric interface will be illustrated with varying parameters.In all our results to be presented, the rotational speed represented by the azimuthal mode indices is found to play a pivotal role in revealing distinct features of the specific spin.By this way, we will be naturally led to the concept of singularity and optical vortices [25].

Problem Formulation
Figure 1(a) sketches a cross-section of a solid cylinder with a fixed radius R , whereas figure 1(b) shows that of a cylindrical hole.Figure 1 , , x y z [8].We emphasize that only rotational waves propagating on the cross-sectional plane of a cylinder are under current investigation.In other words, we do not consider wave propagations along the axial z -direction for simplicity.Hence, we are dealing with two-dimensional wave problems, where all the field variables depend only on either  with n being positive refractive index [10].In this notation, in n refers to the interior over the range 0 r R ≤ ≤ , whereas ex n refers to the exterior over the range R r ≤ < ∞ .Hence, the cylindrical material interface is located at r R = .Whenever necessary, the superscripts in and ex refer henceforth respectively to the interior and exterior.The refractive index contrast can thus be defined to be in ex n n for convenience [20].Therefore, Besides, the exterior is vacuum in figure 1(a), whereas the interior is vacuum in figure 1(b).
In the absence of electric charges, consider Maxwell's equations ( ) . Both electric field vector and magnetic field vector are real.The properties of vacuum are the electric permittivity 0 0 ε > and magnetic permeability 0 0 µ > [1].All the field variables are assumed to follow the combined phase factor ( ) , where m is the azimuthal mode index, ω is frequency, and t is time.We assume 0 ω > throughout this study for temporally non-attenuating waves, thereby leading to a requirement of an energy absorption from somewhere in order to compensate for the radiation loss [21].
As usual, 0,1, 2,3, m = ⋅⋅⋅ for azimuthal periodicity.For 1, 2, 3, m = − − − ⋅⋅⋅ , all the ensuing formulas will be appropriately understood with a reversal in the rotational direction.Under these assumptions, the normalized electric field vector E and magnetic field vector H are defined through ( ) ( ) When equation ( 1) is plugged into Maxwell's equations, z H is found to be governed by the following Helmholtz equation.
( ) It is appropriate to define the reduced radial coordinate r R ρ ≡ . Hence, 0 1 ρ ≤ ≤ in the interior and 1 ρ ≤ < ∞ in the exterior, whereas 1 ρ = refers to the cylindrical material interface.In addition, the positive size parameter q is defined to be 2 q R π λ , where λ is the wavelength of EM waves.Since Maxwell's equations are linear, the magnetic field can assume the following normalized forms respectively in the interior and exterior.
( ) Hereafter, ( ) m J ⋅ is Bessel function of first kind [26].In the exterior, let us define the following twowave-interaction function in the exterior.
Here, either . In addition, H α are Hankel functions of first and second kinds, thereby implying waves respectively outgoing and incoming in the radial direction [20,26].Therefore, A is a complex asymmetry parameter so that we set A b ia ≡ + with , b a being real.When 0 A ≠ , A denotes the deviation from a perfect balance between outgoing and incoming wave [17].Through ( )

Dispersion and Asymmetry Parameter
Let us define the logarithmic derivative being respectively amplitude and phase functions.
Here, the second term ( ) φ α α on the righthand side is the phase gradient [25].Across 1 ρ = , there are three continuity conditions: ( ) ( ) . Via Equations ( 1) and ( 3), the desired We notice that in is real so that ex should be real as well.Physically speaking, both dielectric media are assumed lossless so that waves are not attenuated and energy is conserved.This dispersion relation stems from setting the two interface impedances equal to each other, whereby the logarithmic derivatives are naturally show up [6,22].With Therefore, ( ) ex ex n J n q aY n q J n q n dJ n q dY n q a d n q d n q in ex a a m n n q = can be readily evaluated.We call a state with such a computed a the "neutral" state [20].For presence solely of energy radiation [8,18,20].
In the presence of both energy radiation and absorption, the particular dispersion relation in ex = in equation ( 6) has been derived for the first time in [17], to the best of the authors' knowledge.By this way, in ex = on resonance does contain a kind of input-output relationship between the incoming and outgoing waves [20].It is made possible through the afore-mentioned exterior boundary value problems [21].Numerically it turns out that a undergoes several sign changes with increasing m .In both cases, a approaches zero as m → ∞ .The difference is that the rate of approach to the limit as m → ∞ is faster in figure 2(a) than that in figure 2(b).Physically speaking, the incoming wave falls in perfect balance with the outgoing wave as the rotational speed goes to infinity.For visual aid, figure 2 contains additional straight lines that connect two values of a between two adjoining neutral states with m and 1 m + .( ) Here, g is a constant depending on the system of units.For equation (10), use is made of the relationships for our particular cylindrical wave.
For Equation (11), the term As a result, , which is proportional to the fictitious magnetic field [8].We remark that 0  Let us examine figure 3 for a solid cylinder and figure 4 for a cylindrical hole, for which the common data is 2 translates into, say, 600 R nm = for 600nm λ = .For instance, a nano-scale cylindrical object is hence under consideration.The first impression upon comparing the five panels in figures 3 and 4 is that both energy density and light spin exhibit relatively larger magnitudes on the vacuum sides in comparison to those on the denser dielectric sides.This simple finding is understandable since EM waves feel more freedom in their excursions in vacuum than in denser dielectric media.

Specific Spin
The indeterminacy in the magnitudes of either W or z S stems of course from the linear property of Maxwell's equations [1].Therefore, the specific transverse light spin z σ (to be henceforth shortened as "specific spin") in the axial z -direction is defined with respect to energy density of electromagnetic waves [3,12].
Here, the denominator ( ) In a similar manner, the conventional degree of polarization ,2 rθ

Π
of second order is defined to be Its first-order cousin ,1 rθ Π can be defined in a similar fashion to be

Π
do not satisfy the electromagnetic duality because of the absence of magnetic fields in their respective definitions [10,11,16].Figure 5 shows that 1 (12).
Therefore, the only chance for 1  α > [26].Equation ( 12) is now examined for its limit as 0 ρ → in the following manner.
( ) Therefore, Here, we employed equation ( 1) in equation ( 13).Besides, we took advantage of the equality α ρ ≡ → .Under this condition, the limit process in equation ( 14) would be independent of in n even for lossy dielectric medium with either complex in n or complex in ε .At the same time, it is remarkable that both cross-sectional electric-field components in r E and in E θ make equal contributions to W as 0 ρ → as can be seen from the denominator of equation ( 14).In terms of Cartesian coordinates, both in x E and in y E make equal contributions to W .In other words, perfect circular polarization prevails as 0 ρ → [10,25].For larger m → ∞ , the story would turn out similar as will be shortly discussed.We notice also in figure 5 that the cylindrical holes treated in figures 5(c) and (d) exhibit rather uniform distributions of in z σ in the interior in comparison to the solid cylinders considered in figures 5(a) and (b).This feature is again in accordance with the more freedom in vacuum than in solid.
In case of a solid cylinder, movie 1 exhibits a continuous transition including that from figure 5(c to the last frame with ( ) the spatial undulations get more vigorous.In this aspect, notice that the range 2 q π > has been studied in Ref. [19].For a proper interpretation of the results in this movie 1, we remark however that the location 1 ρ = refers to an increasing radius with increase in q .It is because 2 q R π λ

Jumps in Specific Spin
For quantitative assessment of the jumps in specific spins across the material interface, let us define This jump is related to spin-orbital interactions (SOIs) [3,9,12].See in particular equation (3) of Ref. [8].This jump z σ ∆ is similar to the polarization rotations occurring with chiral materials [10].Whether spin flips or sign changes in z σ take place across the material interface is determined by examining the sign of ( ) ( ) [bliokh15].Numerically, it turns out from examining all the parameter combinations that ( ) ( ) as expected, since both media are dielectric and they are of the same character.There is hence no spin flips across material interfaces in our study.We expect that spin flips can take place for cases, say, with metals in the interior and vacuum in the exterior.In this regard, we find that the discontinuity across a material interface can be made to disappear by metasurface engineering [27].

Discussions
The transverse-electric (TE) wave is also admissible as long as energy supply is maintained from the radial far field.Since the TE wave is demanding more of such energy supply, it is harder to be established than the TM wave.Therefore, we have focused solely on TM waves in this study.In the presence of both energy radiation and absorption, a TE wave with non-zero field variables ( ) 6).Likewise, the factor ex in n n for a TM wave on the left-hand side of equation ( 9) should be replaced by its inverse in ex n n for a TE wave.We hope to elaborate on the TE wave in the near future.
Let us make a brief discussion on Poynting vectors for our cylindrical waves [17].The energy flow of electromagnetic waves is described by Poynting vector ( ) , , ex r P = .Therefore, the sole non-zero component is P θ in the angular direction, which is not of much interest.
We have seen optical vortices at the cylindrical axis from figures 5 and 6.In this regard, notice that the energy density vanishes or 0 in W → as 0 ρ → as long as 2 m ≥ as seen from figures 3(a) and 4(a).Therefore, there exist only a few photons near the cylindrical axis, for which we hope to work out quantum optical formulation as well.For space reasons, polarizations are not discussed in this study.See some results in Ref. [17].As discussed for equation (12), the specific spin z σ is similar to the conventional degrees of electric-field polarization, namely, ,2 rθ Π and ,1 rθ Π , but they are not equal [3,4].In addition, our numerical results show that the effectiveness in achieving circular polarization increases with increasing azimuthal mode index as m → ∞ and decreasing cylinder's radius as 0 q → [10].
In the case of our cylindrical EM waves, the energy supply from the radial far field could eventually lead to axial wave propagations, which would complicate but enrich the dynamics under considerations [9,15].As regards equation (11), we should remark that light spin would not be proportional to the fictitious magnetic field, if  [8,18].The first case of material losses may be incurred by realistic metals or lossy dielectric media.The second case of axial propagations in the presence of energy exchange will be investigated in the future study.

Conclusion
In summary, we have examined both energy density and light spin of cylindrical electromagnetic waves.During problem formulation, the presence of both energy absorption and radiation was necessitated for maintaining time-periodic wave propagations without wave attenuations.By this way, we came naturally up with the light spin per energy density, namely, the specific spin.After establishing the bounded property of the magnitude of the specific spin, we have examined various characteristics of the specific spin for two geometries: the solid cylinder and cylindrical hole.In addition, we found that the specific spin is very sensitive to the size parameter, thus pointing out the peculiarity of nano-scale cylindrical objects.All the analytical tools we have developed in this study would serve as stepping stones on which we could build more delicate formulas as problem complexities increase and hence solutions to Maxwell's equations get harder to be obtained.
(c) displays a transverse-magnetic (TM) wave with its non-zero field components ( )

Figure 1 :
Figure 1: (a) A solid cylinder being optically denser than the exterior.(b) A cylindrical hole being optically rarer than the exterior.(c) A transverse magnetic (TM) wave under this study with non-zero field components.With non-magnetic dielectric media assumed throughout, figure 1 displays two different combinations optical media depending on the relative dielectric constant ε .All the dielectric media are assumed lossless in this study so that we can set 2 n ε ≡with n being positive refractive index[10].In this notation, in n refers to the interior over the range 0 r R ≤ ≤ , whereas ex n refers to the exterior over the range R r ≤ < ∞ .Hence, the cylindrical material interface is located at r R = .Whenever necessary, the superscripts in and ex refer henceforth respectively to the interior and exterior.The refractive index contrast can thus be defined to be

(
left-hand side of equation (8) is real for α being real, we should take 0 b = on the right-hand side.As a result, multiple of the angle π according to equation(5), which leads in turn to a phase gradient ( )d d φ α α being ill-defined.Going back to equation (6), in ex = takes the following linear form with respect to a .

Figure 2 :
Figure 2: The asymmetry parameter A ia = with varying 0 m ≤ ∈ .(a) For a solid cylinder, and (b) for a cylindrical hole.Note the scale change ( ) 0.25 sgn a a a ≡ in (b).The arrows indicate the direction of increasing m .

Figure 2
Figure 2 plots the asymmetry parameter a with varying m as indicated by several integers for 0,1, 2,3, m = ⋅⋅⋅ .As indicated in a box for ( ) , in ex n n , a solid cylinder is considered in figure 2(a), whereas a cylindrical hole is examined in figure 2(b).Numerically it turns out that a undergoes several sign changes with increasing m .In both cases, a approaches zero as m → ∞ .The difference is that the rate of approach to the limit as m → ∞ is faster in figure2(a) than that in figure2(b).Physically speaking, the incoming wave falls in perfect balance with the outgoing wave as the rotational speed goes to infinity.For visual aid, figure2contains additional straight lines that connect two values of a between two adjoining neutral states with m and 1 m + .
vector[8] is non-zero except for the linearly polarized electric field.Besides, the term vertical straight line in brown color throughout this study.The azimuthal mode index is varied over 1, 2, 4,10 m = throughout figures 3-5, where the respective curves are colored in the same way: solid green for 1 m = , broken blue for 2 m = , solid red for 4 m = , and broken black for 10 m = .Notice that each curve in figures 3 and 4 is normalized by the value on the exterior side of the material interface.This fact that ( ) ρ = = can be most clearly seen from figure 4(a) [10].

Figure 4 :
Figure 4: (a,b) Energy density W , and (c) the light spin z S .Although similar to figure 3, a cylindrical hole with 1 2 in ex n n = < = is under consideration.(c) An additional curve for 0 m = is added to the curves in (a), but displayed only in the interior.The effects of the varying m in both figures 3 and 4 are a bit contradictory, depending on the combinations of media.For instance, in W in figure 3(a) undergoes relatively larger variations than ex W for smaller 1, 2, 4 m = .In contrast, for larger 10 m = , ex W in figure 3(a) undergoes relatively larger variations than in W . Turning attention to energy density in both figures 3(a) and 4(a), we find a bright spot with ( ) 0 0

From the data of figures 3
m = .In addition, figures 5(b) and (d) are produced with a smaller size parameter 0.2 q π = .The parameter set is such that ( reversal in the rotational direction from the counter-clockwise to the clockwise directions.

H
This situation takes place only at the cylindrical axis at 1 ρ = , which is optical singularity[7,26].In comparison, a similar situation that → takes place across a material interface in case of helical metamaterials[10].

Figure 5 :
Figure 5: The specific spin In terms of the field variables, the axial transverse magnetic field makes a negligible contribution to the energy density in comparison to the combined electric-field contributions as long as

Consider next figures 5 =
figures 5(a) and (c) experiences sinusoidal sign changes over the radial range, whereas z σ in ) to figure5(a) as q is increased.To this goal, we examined irregular steps.By comparing the first frame of movie 1 with marked in figure 5(b) by the empty circle in blue color, whereas marked in figure 5(b) by the empty circle in brown color.We then find that the minimum either in

Figure 6 :
Figure 6: The jump in the specific spin

Figure 7 :
Figure 7: The jump in the specific spin ,[13].This formula is valid in a time-averaged sense for 0 ω > .Let us evaluate each component of P for our TM wave with nonzero field components ( ) the exterior, recall the asymmetry parameter A ia = with a being real.As a result, as well, thereby leading once more to 0