We presented an operator method to characterise in a systematic way the possible complete orthonormal sets of astigmatic Gaussian modes. The fundamental mode is specified by a
4. Orbital angular momentum of general astigmatic modes
Gaussian function in terms of a complex symmetric 2×2 matrixα. The real part of the matrix specifies the elliptic intensity distribution, and the imaginary part defines the phase distribution. The behaviour of the mode under propagation, as determined by the paraxial wave equation, is given by Eqs. (4.15) and (4.14), in terms of a simplezdependence of the matrix.
A general Gaussian function is also characterised by the requirement that it vanishes when a lowering operator is applied. In two dimensions, this defines two independent lowering op- erators. Higher-order modes can be constructed by repeated application of the corresponding raising operators ˆa†1and ˆa†2, as expressed in Eq. (4.25). Since there are two independent raising operators, the choice of ˆa†1and ˆa†2as linear combinations of the basis set is an inherent degree of freedom, which determines the nature of the set of higher-order modes for a given fundamental mode. This freedom is used by selecting the unitary matrixσ, in one transverse plane. When the linear combinations are basically real,σ is equivalent to a rotation in two dimensions, and the set of modes has the nature of HG modes, which is reminiscent of linear polarisation. In general the choice of the higher-order modes for a given fundamental mode can be represented by a point on the unit sphere. This Hermite-Laguerre sphere is analo- gous to the Poincar´e sphere that represents polarisation. The power of this description is that the astigmatism and the Hermite-Laguerre nature of the modes as a function of the transverse plane is automatically accounted for. Thezdependence of the symmetric matrixα(determin- ing the astigmatism) and the unitary matrixσ (which describes the nature of the modes) are studied in Section 4.4. The OAM of the various sets of modes is analysed in Section 4.5. It can be separated in a term that depends only on the astigmatism and terms that depend also on the selection of the nature of the modes, as determined by the point on the Hermite-Laguerre sphere. In general, astigmatism has a tendency to enhance the OAM.
Simple astigmatism is naturally imposed on a non-astigmatic beam by sending it through an astigmatic lens (or, equivalently, by reflecting it by an astigmatic mirror). Simple astig- matism can be converted into general astigmatism by a second astigmatic lens, with an ori- entation different from the first. An astigmatic lens can be specified by a two-dimensional real symmetric matrixζ, with eigenvalues 1/f1and 1/f2the inverse focal lengths, and with
eigenvectors in the corresponding directions in the transverse plane. The effect of the lens is simply a multiplication of the mode by the factor exp(−ikRζR/2). This is equivalent to a change of the matrixα that determines the astigmatism. The effect of the lens is thatαis replaced byα+ikζ after the lens, which implies thatα1is replaced byα1−kζ. The matrix
σ remains unaffected by the lens. Combined with the effect of free propagation, which is expressed in the simplest way by Eq. (4.39), this information is sufficient to evaluate the variation of all sets of modes as they propagate through a given lens system.
The paraxial wave equation is identical in form to the two-dimensional Schr ¨odinger equa- tion for a free particle. As a consequence, the complete sets of solutions that we derived in this paper are also complete sets of solutions of the Schr ¨odinger equation for a free particle. The fundamental mode corresponds to a wave packet that tumbles upon propagation, and therefore has OAM. Also, the wave packet has a minimal size at two instants of time, corre- sponding to the two focal planes. In general, when the point on the Hermite-Laguerre sphere is not on the equator, the higher-order modes contain vortices, and the corresponding wave packets do as well. The operator method can easily be extended to three dimensions, so that solutions of the three-dimensional Schr ¨odinger equation for a free particle are found.
CHAPTER
5
Vortices in Gaussian light beams
The dynamics of vortices in light beams is complicated in general. Still, it is possible to derive some general properties of vortices in Gaussian beams. We study the vortices that occur naturally in higher-order astigmatic Gaussian beams, and derive a general expression for the positions of the vortex centres. By imposing mathematical vortices on a Gaussian beam, more general statements about the propagation properties of the vortices are obtained. In non-astigmatic Gaussian beams, the trajectory of a canonical vortex represents a dark ray. In a degenerate resonator a vortex mode is obtained in which a dark ray forms a closed trajectory, similar to a light ray in a geometric mode.
Partially based on J. Visser and G. Nienhuis, Proc. of SPIE5736, 126 (2005) and G. Nienhuis and J. Visser, J. Opt. A: Pure Appl. Opt.6, S248 (2004).
5. Vortices in Gaussian light beams
5.1
Introduction
The dynamics of vortices in light beams attracts a lot of attention. Vortices that occur nat- urally in Gaussian beams are studied in the literature both experimentally and theoretically. The trajectory of the vortex centre can be very complicated [41], and the sign of the vortex charge can change [32] under free space propagation. By imposing mathematical vortices on Gaussian beams, some general propagation properties of vortices and their interaction can be obtained [33].
In Section 5.2 we study the vortices that occur naturally in light beams that are in higher- order transverse modes. The vortex centres in astigmatic beams follow a complicated tra- jectory upon propagation. The morphology of a simple vortex can be represented by a point on the morphology sphere [42]. We discuss the relation between the point on the Hermite- Laguerre sphere that characterises the nature of the higher-order modes, and the point on the morphology sphere that represents the morphology of the vortices in the modes.
In Section 5.3 we consider the properties of vortices that are imposed on astigmatic Gaussian beams propagating through free space. We obtain the trajectory of the vortex cen- tre, and show that the charge of a vortex can change sign. We derive conditions under which vortices imposed on a Gaussian beam do not interact.
In Section 5.4 we consider the trajectory of the centre of a canonical vortex imposed on a non-astigmatic Gaussian beam that propagates through a lens guide. In between the lenses the trajectory of the centre of the vortex is a straight line, while at the lenses the direction of propagation changes. Since the intensity of the light beam at the vortex centre vanishes, it represents a dark ray. Inside a degenerate resonator these dark rays form a closed trajectory, in the same way as the light rays of a geometric mode.