Optical Phase Space Research Articles

Electron spin detector for spin-resolved x-ray photoelectron spectroscopy

A low-energy diffuse scattering electron spin detector for spin-resolved x-ray photoelectron spectroscopy (SRXPS) is described. The detector is coupled to a commercial 152-mm mean radius, 150° spherical sector electrostatic electron energy analyzer. Electron optics are described for both the transport of photoelectrons from the energy analyzer exit slit to the gold scattering target, and from the gold target to the spin analyzer electron detectors. The spin detector has an effective Sherman function of 0.078±0.008, with a figure of merit of approximately 4×10−5. The electron optical phase space acceptance EAΩ of the detector is 623 eV mm2 sr. The SRXPS apparatus performance is demonstrated with SRXPS spectra from ferromagnetic Fe and a cobalt-based ferromagnetic glass.

Relativistic aberration of optical phase space

When an optical imaging system is boosted to relativistic velocities, the directions of the incoming light rays converge toward the pole of motion and the images are amplified in that direction and suffer a global comatic aberration. Our analysis uses succinctly the Hamilton–Lie formulation of optics to find the invariants in optical phase space of all Lorentz transformations.

Invariance properties of general astigmatic beams through first-order optical systems

With a new mathematical formulation expressed in terms of the direct product of matrices, the propagation laws for the first-, second-, third-, and fourth-order moments of a general astigmatic beam are derived. It is emphasized that the possibility of finding invariants for propagation through linear optical systems results from the symplectic structure of optical phase space. Invariant quantities for the propagation of general astigmatic beams are defined.

Refracting surfaces between fibers

I characterize the transformation of optical phase space that is due to aligned elliptic index-profile fibers and axially symmetric refracting interfaces between two such media. These include the homogeneous medium as a particular case. I treat the phenomenon in phase space by working with the transformation that is the root of refraction. This is exact in 4π optics and solved by an implicit equation. By means of its polynomial approximation I obtain the seventh-order Lie–Seidel aberration coefficients. The group composition of such transformations permits the symbolic computation of graded-index lens systems. We apply this in a preliminary analysis of Fourier transformers built out of a fiber with a polynomial interface to a homogeneous medium.

Diffraction-free beams remain diffraction free under all paraxial optical transformations.

We point out that the nondiffracting ${\mathrm{J}}_{0}$ beams of Durnin, Miceli, and Eberly are subgroup-reduced basis functions for the group of paraxial (symplectic, linear) transformations of optical phase space. This allows the action of the group on such and similar beams to be computed through 2\ifmmode\times\else\texttimes\fi{}2 matrix algebra.

The Hamiltonian formulation of optics

Snell’s law of refraction is a conservation law of optical momentum. In infinitesimal form, it leads directly to the Hamilton equations for geometrical optics. The latter guarantee that the transformation of optical phase space produced by propagation through a medium where the index of refraction is continuously differentiable, is canonical. Discontinuous-index systems such as refracting surfaces must be treated through a complementary description to show that they are also canonical. They have recently been shown to factorize into two ‘‘root’’ transformations, each depending on the properties of a single medium, and each canonical. This factorization is applicable to reflection off curved mirrors as well.

Symmetry in Lie optics

Optical systems produce nonlinear canonical transformations in optical phase space. Free propagation in a homogeneous medium has Euclidean symmetry and dynamical algebras under the Poisson-Lie bracket. Refracting interfaces between homogeneous media exhibit invariants; in particular, a spherical surface possesses an so(3) symmetry algebra which allows the recursive computation of its aberration coefficients to arbitrarily high order. We present explicit results for aberration order nine.

The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order

The constituents of a lens system, i.e., slabs of homogeneous transparent material and the refracting surfaces between them, considered to third aberration order, are associated to elements of a nine-parameter aberration group. Three parameters correspond to Gaussian systems and six to group-classified aberrations. The group multiplication operation (through matrix-cum-vector algebra) corresponds to their concatenation, and the linear group action on an eight-dimensional homogeneous space corresponds to the nonlinear action of the system on the optical phase space. This leads to economical computation algorithms that may be extended to aberrating systems of higher order.

Factorization of the phase-space transformation produced by an arbitrary refracting surface

We study the transformation produced on optical phase space by an arbitrary refracting surface. We show that this factorizes into two root canonical transformations, one representing the propagation from a reference plane to the refracting surface in the first medium, and the other, the inverse propagation back to the reference surface in the second medium. This factorization allows for a simple parameterization of the effect of a refracting surface to some aberration order, illustrated for cubic surfaces and Dragt’s Lie-algebraic description of a quartic surface and its Seidel aberration coefficients.