We study the geometric origin of generalized Gouy phases in paraxial optical modes of arbitrary order. We focus on
the specific case of cyclic beam transformations of non-astigmatic vortex beams, thereby, generalizing the well-known
geometric phase shift for first-order beams with orbital angular momentum to modes of arbitrary order. Our method
involves two pairs of bosonic ladder operators, which, analogous to the algebraic description of the quantum-mechanical
harmonic oscillator in two dimensions, connect transverse modes of different order. Rather than studying the geometry of
the infinite-dimensional space of higher-order modes, we focus on the space underlying the ladder operators. We identify
overall phases of the ladder operators, thereby obtaining the phases of all higher-order modes, and show that the variation
of these phases under optical elements and transformations has a geometric interpretation in terms of the other parameters
involved.
We study the effects of rotation on the stability properties of an astigmatic two-mirror cavity. We show that
rotation can both stabilize and destabilize a cavity and investigate the effects of such a rotationally-induced
transition on the spatial structure and the orbital angular momentum of the cavity modes. Our method relies
on the connection between ray and wave optics and is exact within the time-dependent paraxial approximation.
We use an algebraic method to derive explicit expressions for the structure of paraxial modes in a cavity consisting
of astigmatic mirrors. The algebra is based upon the use of ladder operators that raise or lower the mode indices,
when acting on a mode function. We show that the method is also applicable when the mirrors perform a uniform
rotation about their axes. We also find expressions for the orbital angular momentum in these modes.
We propose a quantum theory of rotating light beams and study some of its properties. Such beams are
polychromatic and have either a slowly rotating polarization or a slowly rotating transverse mode pattern. We
show there are, for both cases, three different natural types of modes that qualify as rotating, one of which is a
new type not previously considered. We discuss differences between these three types of rotating modes on the
one hand and non-rotating modes as viewed from a rotating frame of reference on the other, thus resolving some
paradoxes mentioned recently.
The modes in an optical cavity between two astigmatic mirrors have a twisted structure when the mirror axes are
not aligned. We use operator techniques to obtain a full characterization of these modes. The method is exact in
the paraxial limit. The structure of the modes is completely determined by the geometry of the resonator. This
geometry is given by the separation between the mirrors, their radii of curvature, and the relative orientation
of their symmetry axes. The fundamental mode has elliptical Gaussian intensity profiles and the intersections
of a nodal plane with a transverse plane normal to the axis can be ellipses or hyperbolae. The symmetry
axes of the intensity curves and the nodal curves are not aligned. At the mirrors, the higher-order modes have a
Hermite-Gaussian structure. Their analytical form can be generated from the fundamental mode by using raising
operators that generalize the operators that are known in the description of the quantum harmonic oscillator. In
the interior region of the resonator, admixture of Laguerre-Gaussian structures can arise, resulting in vortices.
Light beams rotating about their axis can be created using rotating optical elements. We analyze the properties of rotating beams by expanding the mode function in eigenfunctions of angular momentum. Both the spin angular momentum, arising from the polarization, and
orbital angular momentum, arising from the circulating phase gradient, are considered.
We describe basis sets of general astigmatic modes that are solutions of the paraxial wave equation. The fundamental modes are Gaussians with elliptical shapes of the spot size and elliptical or hyperbolic wave fronts. For a given fundamental mode, higher-order modes can be generated by the repeated application of two raising operators. The nature of the set of higher-order modes, corresponding to the Hermite-Gaussian, Laguerre-Gaussian, or intermediary modes, can be characterized by a point on a sphere, in direct analogy to the representation of polarization on the Poincare sphere. For general astigmatism, even the fundamental mode can carry high values of orbital angular momentum per photon. The additional angular momentum of higher modes as well as the vortex structure of the modes depends both on the degree of astigmatism and on the point on the sphere.
We present analytical and invariant expressions for the steady-state density matrix of atoms in a resonant radiation field with arbitrary intensity and arbitrary elliptical polarization. The field drives the closed dipole transition with arbitrary values of the angular momenta of the ground Jg and excited Je states. These transitions can be classified depending on the structure of the solution and its physical peculiarities.
The exact steady-state solution of the optical Bloch equations for the density matrix of atoms interacting with a monochromatic field is found in a compact analytical form for closed Fg equals F yields Fe equals F + 1 transitions. The obtained solution is valid for arbitrary light ellipticity and intensity. Some applications of this new result are outlined.
The quantum-mechanical transport properties of cold atoms in an optical lattice are described in terms of an analytically tractable model. The atoms are cooled down to the lowest energy band in the periodic optical potential, in the tight- binding limit. The transport mechanisms are coherent tunneling through potential barriers, and momentum recoil at optical pumping. Expressions are obtained for the time-dependent population distributions over the energy eigenstates and over localized states. These expressions are modified in the presence of a dark atomic state, that is uncoupled to the field.
The light force on slow atoms in a light field with arbitrary polarization distribution is explicitly evaluated in the case of an atomic transition J yields J with half-integer J values. The expression of the force is based on an exact analytical expression for the steady-state density matrix of the atoms. The force is naturally separated in contributions resulting from gradients of the field intensity, the polarization ellipticity, and the phase. In addition to a conservative part this force also contains a vortex component. In areas of spatial localization, where the total force vanishes, the atomic motion is governed by a potential, which in general is not harmonic for J greater than 1/2. Simple model of the formation of a one- or two-dimensional spatial atomic grating in a specific light field configuration illustrates the results.
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