either by considering the action of the electric field on the dipoles of a material medium, or from the former sing integration by parts. 3, Equation(8)is usually also accompanied by boundary terms dependent upon the profile of the beam, but these are cancelled out during the integration by parts to leave Eq. (9 )a pure volume Khrapko's paradox arises as both the above expressions are held to be directly true for a plane wave, which is asserted to have no boundaries. However, this leads authors to consequently infer angular momentum densities of either I rXS iE(E*×E) the former being zero for a circularly polarised plane wave and the latter being non-zero. Indeed, the popularity of the former view leads not infrequently to papers stating that a plane wave carries no spin(e. g. Refs. 41, 42) and yet is nevertheless capable of generating rotation on interaction with matter(e. g. Ref. 43) 3.2. Theoretical Analysis It is possible that this controversy has to some extent been fuelled by the abstract nature of a plane wave Extending to infinity in all directions, such edgeless beams obviously cannot be reproduced in the laboraos We shall therefore address this controversy both for a circularly polarised plane wave, and for a realistic be such as may be produced in the laborator For the plane wave, one need go no further than the excellent analysis of Mansuripur, who notes tha e product r x p ranges to oo x0 and hence is indeterminate. To overcome this, he considers four identical plane waves initially propagating at an angle 6 from the z axis, and allows them to converge to form a single plane wave. By means of careful limit-taking procedures applied to the central interference fringe he is able to demonstrate that a plane wave does in fact carry angular momentum with a density in free space of l2 is the z component of the angular momentum density, E is the electric field of the plane wave, Eo is the dielectric constant and f is the frequency of the wave. As the energy density of the wave is 4Eoe- and the energy of a photon of frequency f is hf, this expression corresponds to an angular momentum per photon of hi. in agreement with quantum mechanics. The position that a circularly polarised plane wave carries no angular momentum may therefore be rejected, and we find that correctly applied, both treatments yield an angular momentum distributed evenly throughout the bea. We now turn our attention to a finite beam such as may be employed in the laboratory. For our purposes, it is convenient to consider a beam of somewhat arbitrary profile such as that shown in Fig. 1(a). The expression for the electric and magnetic fields of this beam are e= Eoe-fV1-e-aeifwt-ka)*-iy+-(ir-y) a2(er2/a2-1)d2 (13) e0 Eoe-d 1 (x-10(a()-1)- We calculate the angular momentum densities according to eqs.(10) and(11) and obtain the profiles shown in Fig 1(b)and(c) Under both these expressions, the vast majority of the angular momentum of the beam is accounted for within a few scale lengths of the z axis. If the beam is not truncated then it falls off to zero at infinite radius and hence no edge effects exist and the total angular momentum may be obtained by integrating Eqs.(10)and (11)over infinite volume. Alternatively, if the beam is truncated at a radius of several scale lengths(e.g. r= 3either by considering the action of the electric field on the dipoles of a material medium, or from the former using integration by parts.39, 40 Equation (8) is usually also accompanied by boundary terms dependent upon the profile of the beam, but these are cancelled out during the integration by parts to leave Eq. (9) a pure volume integral. Khrapko’s paradox arises as both the above expressions are held to be directly true for a plane wave, which is asserted to have no boundaries. However, this leads authors to consequently infer angular momentum densities of either l = r × S c 2 (10) or l = Re  iε (E∗ × E) 2ω  , (11) the former being zero for a circularly polarised plane wave and the latter being non-zero. Indeed, the popularity of the former view leads not infrequently to papers stating that a plane wave carries no spin (e.g. Refs. 41, 42), and yet is nevertheless capable of generating rotation on interaction with matter (e.g. Ref. 43). 3.2. Theoretical Analysis It is possible that this controversy has to some extent been fuelled by the abstract nature of a plane wave. Extending to infinity in all directions, such edgeless beams obviously cannot be reproduced in the laboratory. We shall therefore address this controversy both for a circularly polarised plane wave, and for a realistic beam such as may be produced in the laboratory. For the plane wave, one need go no further than the excellent analysis of Mansuripur,44 who notes that the product r × p ranges to ∞ × 0 and hence is indeterminate. To overcome this, he considers four identical plane waves initially propagating at an angle θ from the z axis, and allows them to converge to form a single plane wave. By means of careful limit-taking procedures applied to the central interference fringe he is able to demonstrate that a plane wave does in fact carry angular momentum with a density in free space of lz = 4ε0E2 2πf . (12) lz is the z component of the angular momentum density, E is the electric field of the plane wave, ε0 is the dielectric constant and f is the frequency of the wave. As the energy density of the wave is 4ε0E2 and the energy of a photon of frequency f is hf, this expression corresponds to an angular momentum per photon of ¯h, in agreement with quantum mechanics. The position that a circularly polarised plane wave carries no angular momentum may therefore be rejected, and we find that correctly applied, both treatments yield an angular momentum distributed evenly throughout the beam. We now turn our attention to a finite beam such as may be employed in the laboratory. For our purposes, it is convenient to consider a beam of somewhat arbitrary profile such as that shown in Fig. 1(a). The expression for the electric and magnetic fields of this beam are E = E0e − r 2 d2 q 1 − e − r2 a2 e i(ωt−kz) ( ˆx − iˆy + " 1 k (−ix − y) 1 a 2 ￾ e r 2/a2 − 1
− 2 d 2 !#ˆz) (13) H = r 0 µ0 E0e − r 2 d2 q 1 − e − r2 a2 e i(ωt−kz) ( −iˆx − ˆy − " 1 k (x − iy) 1 a 2 ￾ e r 2/a2 − 1
− 2 d 2 !#ˆz) . (14) We calculate the angular momentum densities according to Eqs. (10) and (11) and obtain the profiles shown in Fig. 1(b) and (c). Under both these expressions, the vast majority of the angular momentum of the beam is accounted for within a few scale lengths of the z axis. If the beam is not truncated, then it falls off to zero at infinite radius, and hence no edge effects exist and the total angular momentum may be obtained by integrating Eqs. (10) and (11) over infinite volume. Alternatively, if the beam is truncated at a radius of several scale lengths (e.g. r = 3d)