Spin annihilations of and spin sifters for transverse electric and transverse magnetic waves in co- and counter-rotations

Summary This study is motivated in part to better understand multiplexing in wireless communications, which employs photons carrying varying angular momenta. In particular, we examine both transverse electric (TE) and transverse magnetic (TM) waves in either co-rotations or counter-rotations. To this goal, we analyze both Poynting-vector flows and orbital and spin parts of the energy flow density for the combined fields. Consequently, we find not only enhancements but also cancellations between the two modes. To our surprise, the photon spins in the azimuthal direction exhibit a complete annihilation for the counter-rotational case even if the intensities of the colliding waves are of different magnitudes. In contrast, the orbital flow density disappears only if the two intensities satisfy a certain ratio. In addition, the concepts of spin sifters and enantiomer sorting are illustrated.

(S1.2) For this co-rotational case, Eq. (S1.1) translate into To emphasize that these gradients take non-zero value only for spatially inhomogeneous fields, we may call these gradients "inhomogeneity gradients". (S1.10) The Hankel functions follow a similar set of rules for derivatives. In addition, we need to pay special attentions to the azimuthally stationary case with 0 m  .   (S1.6)-(S1.8). In similarity to   Im 0 m G   , we notice   Im 0 m K   in the interior, which will be useful for the forthcoming Eq. (S8.24) in evaluating the radial component of the orbital angular momentum.
In addition, we will employ the "helicity"  and "chirality"  .

S2. Energy Density
The energy density defined by Recalling that     Here,  is the chirality defined in Eq. (S1.11). As expected, the azimuthal term With the help of various relations for complex variables, these three are further simplified. (S4.10) Notice the sign change in the expression for Poyn P  The final step is to call upon the pair of solutions (S4.12) By way of the helicity     (S4.14)

S5. Energy Flow Density and Poynting Vector
Via kc   and the Maxwell's equations As a result, the total energy flow density (FD) is identical to the Poynting vector (PV), namely, tot Poyn PP  for dielectric media. This fact holds true to both rotational cases.
Meanwhile, the decomposition of tot P into its orbital FD O P and spin FD S P is easier to handle in the Cartesian coordinates by use of the repeated indices.
In order prove the identity 00 tot O S P P P , let us prove the following identity for the electric field first.
For dielectric media without space charges, we have On the other hand, consider next for complex fields, but by its respective imaginary parts. In addition, we should remark that the divergence-free conditions 0 E   and S18 0 H    have been incorporated in the procedure. In other words, homogeneous dielectric media are assumed without any presence of space charges.

S6. Orbital Flow Density in the Cartesian Coordinates
Now, let us further process the orbital part O P for our particular waves described by We rely on the Maxwell's equations Consider the electric field first by noticing that Here, one is prone to making a grave mistake when setting This expansion make a correction for the above-mentioned mistake in Eq. (S6.2). Let us explain the summation conventions employed here. Any double indices mean a summation over that index as in the Einstein's notation. For the triple (3 times) or quartic (4 times) indices, we employ the summation symbol j  in an explicit way. Therefore, Here, we utilized either Eq. (S1.2) or (S1.4) and the axial-coordinate independence of the electric field.
Similarly, we treat the magnetic field to obtain   (S6.7) Let us go further to separate O P into four parts.
Here, the additional superscripts "TE" and "TM" refer respectively to the TE and TM modes.
We can check a partial validity of these items by finding that the respective sets   ,, ,, x y z f f h appear in the items representing the TE and TM modes, respectively.

S22
Via either Eq. (S1.3) or Eq. (S1.5), the above Eq. (S6.9) can be expressed solely in terms of the two axial components   , zz fh. (S6.10) We find that the imaginary unit do not explicitly appear in these four defining equations due to self-cancellations. As a consequence, Eqs. (S6.10) holds true for both rotational cases.
In the Cartesian coordinates, Eq. (S6.10) lends itself easily to a vector form. (S6.13) Let us introduce another differential vector operator. For this purpose, we have expressed the orbital FD in another vector form in (S6.15).

S7. Vector Laplacians
As another example of the operators on vectors, consider the following vector Laplacian for a generic vector V .
in a short-hand notation for the Cartesian Here, we employed a short-hand notation for a pair of angles (S7.7) This step works fine, since 2 x V  and the likes are the scalar Laplacians, and therefore they can be written in polar coordinates without incurring any errors. (S7.11) While proceeding, we need to pay a special attention to the factors like (S7.12) We take derivatives of the above with respect to the azimuthal angle to obtain Taking the azimuthal derivatives once more, (S7.14) S28 Here, there are two extra terms. Both

S8. Orbital Flow Density for the Rotating Cases
We remark that both rotating cases cannot be resolved in the Cartesian coordinates in a proper way. In order to differentiate between the co-and counter-rotating cases, we should resort hence to the polar coordinates. To this goal, let us treat Eq. (S6.8) given in the Cartesian coordinates into the polar coordinates via Eq. (S7.5).
Hence, we proceed with Eq. (S8.5) while keeping the order of differentiations to obtain It is simplified further. (S8.25) Here, the superscripts "  " and "  " refer respectively to the clockwise and counter-clockwise rotations.
In comparison, for the counter-rotational case, the azimuthal components of the orbital FD is found as follows, when taking the respective imaginary parts.

S9. Spin Vector
The spin vector is given by   Here, we utilized Eq. (S9.10) for a genetic complex variable A and q . Therefore, for the counter-rotational case, we obtain the following with the chirality  defined in Eq. (S1.11).