Scalar beam


In classical optics the term beam is a synonym for solution of the paraxial wave equation. Such solution is associated with vector field and even the so called scalar solution is often thought of as a single-component vector. Hence the term beam in the classical optics is inevitably related to the term polarization. The only reason to introduce the term scalar beam is to reject any association with the classical “polarization”.

In quantum mechanics the term polarized beam means beam that can not be more filtered by Stern-Gerlach filter. One can speak about polarized beam only in context of some filter. The particles of a polarized beam are in a basis state of this filter (Feynman lectures, volume three).

The scalar beam can be considered as a blackbody radiation, which is filtered to form a parallel beam of isomomentum particles. Each beam of particles deviates from the parallel one. The beam diffraction is not essential for the splitting effect of one Stern-Gerlach filter. The diffraction is important to optimize the filter resolution only.

The photon from one idealized scalar beam is determined uniquely by the propagation vector $ \mathbf{n}(\theta, \varphi)$ and by the scalar variable wavelength $ \lambda$ . From $ \mathbf{n}$ and $ \lambda$ there are derived other photon characteristics: the photon field energy $ E_{\lambda} = \mathit{hc}/\lambda$ , the wave vector $ \mathbf{k} = \mathbf{n} / \lambdabar$ , the photon 3-momentum $ \mathbf{p} = \hbar\mathbf{k}$ and 4- momentum $ p_{\mu} = (cE_{\lambda}, \mathbf{p})$ , etc.

In terms of the classical optics, the scalar beam can be thought of as scalar monochromatic plane wave traveling along $ \mathbf{n}$ and diffracted by circular aperture with radius much greater than wavelength $ \lambda$ . In the aperture plane there is not radial phase variation and the Poynting vector is parallel to $ \mathbf{n}$ . The later means that all photons have parallel 3-momentum in this plane, i.e. this is an isomomentum plane.

In terms of the laser optics, the randomly polarized fundamental resonator mode is a Gaussian scalar beam whose isomomentum plane is the waist plane. (The higher transfer modes are not scalar beams.)

Any kind of monochromatic light beam, describable by solution of the paraxial scalar wave equation is also considered as scalar beam, if this solution has an isomomentum plane.

By analogy with the scalar beam, the isomomentum plane can be defined also for beam describable by vector or spinor field. The isomomentum plane is transformed by lens to isomomentum plane too and both are object/image conjugated planes. When a beam with isomomentum plane passes through transparent optically isotropic slab, the isomomentum plane of the passed- through beam is shifted by the “shift of the focus”, known from the geometrical optics. For optically anisotropic crystal slab the “shift of the focus” is determined if the beam passes along the optical axis.


May 24, 2006
Author: T Kirilov