Swings and Roundabouts

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Here, we take the opportunity to explore another such analogy: the connection between polarization—states of elliptic polarization, as parametrized by Stokes parameters and the Poincaré sphere—and the analogous representations of high-order Gaussian laser modes, especially the celebrated Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) mode sets.

The similarity between the Poincaré sphere parametrization of linear, circular and elliptic states of polarization, and HG, LG as well as the less familiar generalized Hermite–Laguerre–Gaussian (GG) modes [2–4], has been much explored over the last 25 years, following the important observation of the ‘equivalence’ of the Poincaré sphere for polarization and Gaussian modes of mode order unity by Padgett & Courtial [5], as shown in figure 1. We will describe how this analogy emerges naturally, from interpreting polarization in terms of the Hamiltonian mechanics of an isotropic two-dimensional oscillator, and Gaussian modes from its canonical quantization in terms of operators . The notion of angular momentum, in the sense of both the spin angular momentum encapsulated by the third Stokes parameter S3 and the orbital angular momentum operator of which the LG modes are eigenfunctions, plays a central role in this picture.Illustration of the Poincaré spheres for elliptic polarization and of Gaussian beams of mode order 1. (a) Polarization ellipses. The ellipses have a right-handed sense in the northern hemisphere, left-handed in the southern hemisphere, with circular polarizations at the poles and linear polarization on the equator. The axis angle of the ellipse is half the azimuth angle on the sphere. (b) Gaussian beam sphere. Rotated HG modes occur on the equator instead of linear polarizations, and LG vortex modes (of positive or negative sign) occur at the poles instead of linear polarization. The analogues of elliptical polarization have a single vortex on axis with an elliptical core. (Online version in colour.)Many aspects of the story we tell have been described before in some detail, especially in [2–4,8–10], although we draw stronger mathematical analogies between the classical Hamiltonian structure of the Poincaré sphere for polarization, its formal canonical quantization for Gaussian beams and the semiclassical relationship between the two. The ‘swings and roundabouts’ nature of harmonic oscillator orbits underlies everything, especially the angular-momentum-carrying nature of circular orbits and LG modes [11]. Our exposition will be pedagogical, with the aim of making the material accessible to new entrants to the field, as well as giving new insight to more seasoned researchers.It is tempting, but potentially misleading, to think of optical angular momentum of structured Gaussian beams directly in terms of three-dimensional quantum spin. In paraxial beams (as considered here), there is only one possible direction of spin or orbital angular momentum, namely the propagation direction, whereas more general quantum spins may be turning about any axis in three dimensions. The analogy instead lies with the structure of different bases of representation. For polarization, this is any pair of orthogonal elliptic polarization states (e.g. linear horizontal and vertical, or right- and left-handed circular polarizations), and these basis states are parametrized by the Poincaré sphere 12]. This sphere is analogous to the Bloch sphere for quantum spin 1/2 because, for light beams with a fixed direction of propagation, the electric field must be transverse, and hence a complex superposition of left and right circular polarizations, or equivalently, vibrations in the x and y directions. Any pair of orthogonal polarizations (i.e. complex two-dimensional Jones vectors) are antipodal on the Poincaré sphere, as we will discuss in §2.For orbital angular momentum, this will be described using bases of Gaussian laser modes—especially the HG and LG basis sets—whose linear relationship is similar to quantum spin bases with different directions of rotation. All of the discussion of Gaussian modes will be restricted to their amplitude distribution in the focal plane (z=0), so a fundamental Gaussian beam has amplitude , where w0 represents the waist width of the beam [13], and is normalized (its square, integrated over the plane, gives unity). This Gaussian has the same functional form as the ground state of a two-dimensional quantum harmonic oscillator, which is justified physically [8,13–15] in terms of the curved mirrors in the laser cavity having the effect on the paraxially propagating wave, in the focal plane, of a harmonic potential.

Most laser cavities have residual astigmatism, breaking the cavity’s pure axial symmetry; the HG modes (TEM modes) are higher-order modes of such cavities, given by

1.1

where Hm,Hn denote Hermite polynomials [16], and m,n are non-negative integers 0,1,2,… (the fundamental Gaussian being the case m=n=0). HG modes are characterized by a nodal ‘grid’ as seen in figure 2a,b, and the function can be completely factorized into two functions, one depending on x (indexed by m) and one on y (indexed by n), and N=m+n is the mode order. For mode order N, there are N+1 modes, for which (m,n)=(N,0),(N−1,1),…,(0,N). The set of HGmn modes is orthonormal (with respect to integration over the plane with uniform weight), and on propagation the modes maintain the same intensity pattern, even to the far field: the Fourier transform of an HG beam is functionally the same as (1.1). On propagation, the modes acquire an (N+1)-dependent Gouy phase factor; the intensity pattern of superpositions of modes with different N changes on propagation, although it does not for superpositions with the same N.

Open in a separate windowFigue 2.

Illustrations of LG and HG beams in the focal plane, and upon propagation. (a) Intensity of HG21 in the focal plane. (b) As the HG21 beam propagates, it spreads while maintaining the intensity pattern. The orthogonal sheets represent the zeros of the HG beam, which are always at the positions of the zeros of the scaled Hermite polynomials. (c) Intensity of LG11 in the focal plane. (d) As the LG11 beam propagates from its focal plane, it spreads maintaining the intensity pattern, but now each equiphase surface, such as the one represented by the grey surface, swirls around on propagation.

The study of optical orbital angular momentum is mainly based around the LG modes [11], which are expressed in plane polar coordinates R,ϕ in the waist plane as 1.2.

whose radial dependence is determined by the associated Laguerre polynomial  [16], p=0,1,2,… and  is a positive or negative integer 0,±1,±2,…. LG modes factorize into an R-dependent function times ; this latter is an eigenfunction of the orbital angular momentum operator −i∂ϕ=−i(x∂y−y∂x) with eigenvalue . LG modes also occur as modes of laser cavities with mirrors with non-negligible spherical aberration; due to residual astigmatism or localized cavity defects, there is a coupling of both signs of angular momentum, so the resulting LG cavity modes occur as the real and imaginary parts of (1.2), and so do not carry a sense of right- or left-handed angular momentum.

LG modes also form a complete basis, have the same functional form as their Fourier transform and maintain their intensity pattern on propagation, as shown for the example LG11 in figure 2c,d. The phase swirls in such a way that the z component of orbital angular momentum of the transverse beam is preserved, in addition to the Gouy phase. An LG mode has mode order N=||+2p: for each N there are again N+1 modes where (,p)=(−N,0),(−N+2,1),…,(N−2,1),(+N,0). Each LG mode with mode order N can be expressed as a superposition of the N+1 HG modes of the same mode order, and vice versa [2]; a major aim of this paper is to explore this connection in detail.

Padgett & Courtial [5] observed that any superposition of Gaussian beams of mode order N=1 can be represented on a sphere, analogous to the Poincaré sphere of polarization, as represented in figure 1b: the circular modes LG±1,0 occur at the poles, and linear modes HG10, HG01 with any orientation of Cartesian axes occur around the equator; intermediate ‘elliptic’ states occur at other latitudes of the sphere. It is natural to ask what happens to this sphere for different mode orders, and what is the connection with quantum spin and the Bloch sphere.1

In answering this question, we will make much of the two-dimensional isotropic harmonic oscillator, using both classical Hamiltonian mechanics and its canonical quantization in quantum mechanics. As all classical and quantum properties of this system are well known, we simply have to identify these with the known properties of elliptic polarization and Gaussian beams. It is important to note that, in this work, the classical picture is that of elliptic ‘rays’ parametrized by the Poincaré sphere, and the wave mechanical one the Gaussian beams (as eigenfunctions of certain natural operators). Although the language (and indeed the two-dimensional harmonic oscillator system) is suggestive of photons and quantum optics, all our classical, semiclassical and quantum discussion is between rays and waves.

In the next section, we will consider the Stokes parameters and Poincaré sphere, discussing historical approaches and the formulation in terms of the Hamiltonian mechanics of a two-dimensional harmonic oscillator. LG, HG and GG modes are considered in §3 as the eigenstates of operators naturally arising from the canonical quantization of the oscillator, and the connection is strengthened in §4, in which a semiclassical picture relating the ‘classical’ polarization sphere and the ‘quantum’ Gaussian mode sphere is described. Throughout, the rectilinear ‘swinging’ of linear polarization (and HG modes) contrasts with the ‘roundabout’ motion of circular polarization (and LG modes).