The angular momentum of electrons, photons and other quantum particles can be decomposed into spin angular momentum (SAM) and orbital angular momentum (OAM). Spin, or for photons polarization, is known as intrinsic , as it is independent of any spatial origin. OAM can be further decomposed into intrinsic OAM, or IOAM, and extrinsic OAM, or EOAM. [ONeil, Berry*, Bliokh, etc] IOAM is defined relative to the centroid (center of mass or of energy) of a quantum wave function or wave packet; EOAM is associated with the trajectory of the centroid of a wave packet, and is relative to a chosen spatial origin. For example, the OAM of an electronic energy eigenstate in an atom is intrinsic, as is that of a helical phase vortex within an electron beam [BenMcM] or photon beam [Allen?]. Berry showed that these IOAMs are, like spin, independent of any spatial origin.* curved path defined by a helically coiled optical fiber.

Spin-orbit interaction (SOI) involves the interactions of SAM and OAM, for example spin-IOAM interaction as in Russell-Saunders L-S coupling in a single-electron atom, where the interaction Hamiltonian is ((including the relativistic Thomas precession))

where is the gradient of potential energy, the usual OAM and SAM three-dimensional vector operators. A well-known example of spin-EOAM interaction is seen in the precession of the linear polarization vector of photons traveling in a coiled single-mode optical fiber, wherein the photons are forced to follow a non-straight path[Tomita Chiao]. Both of these effects are deeply connected to Berry’s geometric phase*** or gauge potential, as shown for the intrinsic electron case by Mathur [] and for the extrinsic photon case by Wu and Chiao [].

For photons, a progression from the dominance of EOAM to IOAM can be seen in the following examples: Spin-EOAM interaction is seen in a narrow collimated light beam as it travels in a helical path by multiple internal reflections inside a large-diameter (10 mm?) glass cylinder [bliokh], mimicking the coiled-fiber case. For a cylinder geometry comprised of a highly multimode optical fiber with core diameter xx microns, spin-EOAM interaction is observed when the light in the many modes interfere coherently, creating a speckle pattern, which is seen to rotate intact around the fiber axis with a positive of negative angles depending upon the circular polarization (helicity) of the light [Zeldovich]. This phenomenon case be adequately modeled using a ray-tracing or trajectory approximation, enforcing its close connection with EOAM [Dooghin]. At the same time, it has hallmarks of IOAM in that the individual stationary modes that are interfering can each be described as carrying IOAM relative to its own centroid at the fiber’s center. Unlike in the single-mode fiber, linear polarization is not preserved in a highly multimode one, so linear-polarization rotation cannot be observed. [Galvez papers are not in fiber?]

The limit of purely intrinsic OAM is reached in a straight, small-diameter fiber, where both spatial-pattern rotation and linear-polarization rotation are predicted to result from spin-IOAM interaction [cody]. Such a fiber supports only a small number of modes, and these can be individually excited, in contrast to in a highly multimode fiber. It is also in contrast to a single-mode fiber, which can support only the zero-IOAM mode and so spatial rotation cannot be observed. The theory makes highly specific predictions about the relation between spatial and polarization rotations: 1. The rotation angles should be linear with fiber length, 2. The spatial rotation rate should be an integer multiple of the polarization rotation rate, depending on the value of the IOAM. 3. The spatial rotation rate should be opposite and symmetric for left- and right-handed circular polarization, and 4. The polarization rotation rate should be opposite and symmetric for opposite OAM values of the same magnitude [cody]. We present results of an experiment that confirms all of these predictions, providing strong evidence for the existence of purely intrinsic SOI of light.

A further purpose of this study is to emphasize the deep analogy between spin-IOAM interaction in electrons and that in photons. The theory study showed that for either electrons or photons traveling in a few-mode cylindrical waveguide, the SOI Hamiltonian can be put into the common form,

Fat cylinder MM fiber

Is RS coupling really intrinsic? ONeil says “Orbital angular momentum is intrinsic only when the interaction with matter is about an axis where there is no net transverse momentum.”

[safely paraphrasing Bliokh Zayats p797]

WikiP: https://en.wikipedia.org/wiki/Spin%E2%80%93orbit_interaction

*** geometrical phase was Berry’s term but is not used as often