The orbital angular momentum of light (OAM) is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. OAM can be split into two types. The internal OAM is an origin-independent angular momentum of a light beam that can be associated with a helical or twisted wavefront. The external OAM is the origin-dependent angular momentum that can be obtained as cross product of the light beam position (center of the beam) and its total linear momentum. While widely used in laser optics, there is no unique decomposition of spin and orbital angular momentum of light.
Concept
A beam of light carries a linear momentum P {\displaystyle \mathbf {P} } , and hence it can be also attributed an external angular momentum L e = r × P {\displaystyle \mathbf {L} _{e}=\mathbf {r} \times \mathbf {P} } . This external angular momentum depends on the choice of the origin of the coordinate system. If one chooses the origin at the beam axis and the beam is cylindrically symmetric (at least in its momentum distribution), the external angular momentum will vanish. The external angular momentum is a form of OAM, because it is unrelated to polarization and depends on the spatial distribution of the optical field (E). A more interesting example of OAM is the internal OAM appearing when a paraxial light beam is in a so-called "helical mode". Helical modes of the electromagnetic field are characterized by a wavefront that is shaped as a helix, with an optical vortex in the center, at the beam axis (see figure). If the phase varies around the axis of such a wave, it carries orbital angular momentum. In the figure to the right, the first column shows the beam wavefront shape. The second column is the optical phase distribution in a beam cross-section, shown in false colors. The third column is the light intensity distribution in a beam cross-section (with a dark vortex core at the center). The helical modes are characterized by an integer number m {\displaystyle m} , positive or negative. If m = 0 {\displaystyle m=0} , the mode is not helical and the wavefronts are multiple disconnected surfaces, for example, a sequence of parallel planes (from which the name "plane wave"). If m = ± 1 {\displaystyle m=\pm 1} , the handedness determined by the sign of m {\displaystyle m} , the wavefront is shaped as a single helical surface, with a step length equal to the wavelength λ {\displaystyle \lambda } . If | m | ⩾ 2 {\displaystyle |m|\geqslant 2} , the wavefront is composed of | m | {\displaystyle |m|} distinct but intertwined helices, with the step length of each helix surface equal to | m | λ {\displaystyle |m|\lambda } , and a handedness given by the sign of m {\displaystyle m} . The integer m {\displaystyle m} is also the so-called "topological charge" of the optical vortex. Light beams that are in a helical mode carry nonzero OAM. As an example, any Laguerre-Gaussian mode with rotational mode number l ≠ 0 {\displaystyle l\neq 0} has such a helical wavefront.
Formulation The classical expression of the orbital angular momentum is the following:
L = ϵ 0 ∑ i = x , y , z ∫ ( E i ( r × ∇ ) A i ) d 3 r , {\displaystyle \mathbf {L} =\epsilon _{0}\sum _{i=x,y,z}\int \left(E^{i}\left(\mathbf {r} \times {\boldsymbol {\nabla }}\right)A^{i}\right)d^{3}\mathbf {r} ,}
where E {\displaystyle \mathbf {E} } and A {\displaystyle \mathbf {A} } are the electric field and the vector potential, respectively, ϵ 0 {\displaystyle \epsilon _{0}} is the vacuum permittivity and we are using SI units. The i {\displaystyle i} -superscripted symbols denote the cartesian components of the corresponding vectors. For a monochromatic wave this expression can be transformed into the following one:
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