Twisted Photons

Applications of Light with Orbital Angular Momentum

Juan P. Torres, Lluis Torner, Juan P. Torres, Lluis Torner

• Publisher: Wiley-VCH
• Year: 2011
• ISBN:
  9783527635375
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Optics & Light

This book deals with applications in several areas of science and technology that make use of light which carries orbital angular momentum. In most practical scenarios, the angular momentum can be decomposed into two independent contributions: the spin angular momentum and the orbital angular momentum. The orbital contribution affords a fundamentally new degree of freedom, with fascinating and wide-spread applications. Unlike spin angular momentum, which is associated with the polarization of light, the orbital angular momentum arises as a consequence of the spatial distribution of the intensity and phase of an optical field, even down to the single photon limit. Researchers have begun to appreciate its implications for our understanding of the ways in which light and matter can interact, and its practical potential in different areas of science and technology.

1

The Orbital Angular Momentum of Light: An Introduction

Les Allen and Miles Padgett

1.1 Introduction

Most physicists know that polarized light is associated with the spin angular momentum of the photon. It is almost certainly true that the idea of orbital angular momentum is a good deal less understood. Perhaps the simplest and most obvious display of both the spin and orbital angular momentum of light beams comes from an examination of the ratio of their angular momentum to their energy.

For an idealized, circularly-polarized plane wave, the spin angular momentum is given by Jz = N

and the energy by W = N

ω, where N is the number of photons. The angular momentum to energy ratio is thus,

1.1

In fact the ratio in Eq. (1.1) is derivable from classical electromagnetism without any need to invoke the concept of a photon or any other quantum phenomenon [1].

A slightly more general result for elliptically polarized light, characterized by − 1 ≤ σ ≤ + 1, (with σ = ± 1 for left- and right-handed circularly polarized light respectively and σ = 0 for linearly polarized light) is given by

1.2

We can show for a light beam which has an l-dependent azimuthal phase angle such that the field amplitude is given by u(x, y, z, ϕ) = u0(x, y, z)e−ikze+ilϕ, that Eq. (1.2) becomes [2]

1.3

Here

σ describes the spin angular momentum per photon, while l

describes the orbital angular momentum per photon. In the absence of the phase term exp(ilϕ), Eq. (1.3) would be the usual plane wave ratio of spin angular momentum divided by energy, namely,

σ/

ω or

σ per photon.

It transpires that this simple result is true both in the limit of the paraxial approximation and for fields described by a rigorous and unapproximated application of Maxwell's equations [3]. In the paraxial approximation, other than assuming that u(x, y, z) is normalizable and leads to a finite energy in the beam, no assumption has been made about the form of the distribution. In other words even for σ = 0, when the light is linearly polarized, there remains an angular momentum related to the spatial properties of the beam and dependent on l.

The fact that the simple paraxial result, Eq. (1.3), is fully justified by rigorous theory [4] enables a number of essentially simple conclusions to be drawn. The paraxial fields appropriate for linearly polarized light are

1.4

and

1.5

These allow evaluation of the time-averaged Poynting vector, ε 0 E × B, namely,

1.6

For a field such as u(r, ϕ, z) = u0(r, z)e+ilϕ the ϕ-component of linear momentum density is

1.7

while its cross product with r gives an angular momentum density of magnitude jz = ε 0ω

|u|2. The energy density of such a beam is

1.8

Thus,

When the angular momentum density is integrated over the x–y plane, the ratio of angular momentum to energy per unit length of the beam is simply,

1.9

The same straightforward calculation for fields that include polarization, again produces Eq. (1.3), but it is now for physically realizable fields and not just plane wave fields of infinite extent.

The earliest work on the orbital angular momentum of light beams took an LG (Laguerre–Gaussian) mode as the most easily available source of light possessing an azimuthal phase. This amplitude distribution, up, l, has the requisite exp(ilϕ) term and is now well known. It readily follows for such a distribution that the linear momentum density is [2]

1.10

and the cross product with r gives the angular momentum density,

1.11

The expression for linear momentum p, (Eq. (1.9)), shows that at a constant radius, r, the Poynting vector maps out a spiral path of well-defined pitch,

1.12

However, such a picture is misleading as it ignores the radial component of the Poynting vector and, hence, the spreading of the beam upon propagation [5]. For constant r(z)/w(z), the angle of rotation, θ, of the Poynting vector from the beam waist at z = 0 is

1.13

For a p = 0 mode, for which the intensity distribution is a single ring, the radius of the maximum amplitude in the mode is given by

1.14

and so for p = 0,

≠ 0, it follows that

which, surprisingly, is independent of

. Rather than describing a multiturn spiral as one might have presumed, the Poynting vector rotates only by π/2 either side of the beam waist as the light propagates to the far field. Perhaps even more surprisingly, the locus of the vector is simply a straight line at an angle to the axis of the beam [6, 7]. Note that the arctan term is simply proportional to the Gouy phase of the Gaussian beam and that, in free space, the Poynting vector is at all points parallel to the wavevector.

Simple though these results are, in hindsight, they were not known until the early 1990s. Their application to a number of conceptually straightforward experiments enables simple comparisons to be made, at least in the paraxial regime, between the behavior of spin and orbital angular momenta and enables the observation of a number of phenomena to be elucidated. This phenomenology provides much of the basis for the exploration and exploitation of the current understanding of the subject outlined in later chapters of this book. Although everything may be justified formally using a quantum approach, there is, outside of entanglement, little need to leave this classical formulation. In the nonparaxial case, the separation of spin and orbital angular momentum is more complicated [4, 8–10].

The use of the flow of angular momentum flux across a surface, rather than angular momentum density, allows the separation of the spin and orbital angular momentum parts in a gauge invariant way. This holds beyond the paraxial approach but confirms the simple values obtained for the ratio of angular momentum to energy [11].

1.2 The Phenomenology of Orbital Angular Momentum

Simple comparisons of the behavior of spin and orbital angular momenta in different situations prove to be a fruitful way to demonstrate their properties. First, however, we need to distinguish the general structures of light emitted by a laser and also its properties when converted to, for instance, an LG beam. Laser beams usually have spherical wave fronts while the azimuthal phase leads to beams with l intertwined helical wave fronts (Figure 1.1). The LG beam is not the only example of a helical wave front; Bessel beams [12], Mathieu beams [13], and Ince–Gaussian beams [14] can also carry orbital angular momentum. In all cases, the interference of these helical wave fronts with a plane wave gives rise to characteristic spiral interference fringes [15–17].

Figure 1.1 The helical wave fronts characterized by an azimuthal phase term (l = 1) and the associated Poynting vector, the azimuthal component of which gives rise to an orbital angular momentum.

The production of a pure, high-order LG mode from a laser beam was first achieved using a mode convertor based on cylindrical lenses [18]. Although the details are interesting, they need not concern us here, as an approach based on simple holograms achieves a similar beam ...