Interpreting angular momentum transfer between electromagnetic multipoles using vector spherical harmonics

Roger Grinter; Garth A. Jones

doi:10.1364/OL.43.000367

Optics Letters
Vol. 43,
Issue 3,
pp. 367-370
(2018)
•https://doi.org/10.1364/OL.43.000367

Interpreting angular momentum transfer between electromagnetic multipoles using vector spherical harmonics

Roger Grinter and Garth A. Jones

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Roger Grinter and Garth A. Jones, "Interpreting angular momentum transfer between electromagnetic multipoles using vector spherical harmonics," Opt. Lett. 43, 367-370 (2018)

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Abstract

The transfer of angular momentum between a quadrupole emitter and a dipole acceptor is investigated theoretically. Vector spherical harmonics are used to describe the angular part of the field of the mediating photon. Analytical results are presented for predicting angular momentum transfer between the emitter and absorber within a quantum electrodynamical framework. We interpret the allowability of such a process, which appears to violate conservation of angular momentum, in terms of the breakdown of the isotropy of space at the point of photon absorption (detection). That is, collapse of the wavefunction results in loss of all angular momentum information. This is consistent with Noether’s Theorem and demystifies some common misconceptions about the nature of the photon. The results have implications for interpreting the detection of photons from multipole sources and offers insight into limits on information that can be extracted from quantum measurements in photonic systems.

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Equations (11)

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	$E_{1,0, + 1}$	$E_{1,0, 0}$	$E_{1,0, - 1}$
$E_{1,0, + 1}^{*}$	$- 1 / 4 π$	0	0
$E_{1,0, 0}^{*}$	0	$- 1 / 4 π$	0
$E_{1,0, - 1}^{*}$	0	0	$- 1 / 4 π$

	$E_{2,1, + 2}$	$E_{2,1, + 1}$	$E_{2,1, 0}$	$E_{2,1, - 1}$	$E_{2,1, - 2}$
$E_{2,1, + 2}^{*}$	$+ \frac{3}{8 π} \sin^{2} ϑ$	$- \frac{3}{8 π} e^{- i ϕ} \sin ϑ \cos ϑ$	$+ \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- 2 i ϕ} \sin^{2} ϑ$	0	0
$E_{2,1, + 1}$	$- \frac{3}{8 π} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{3}{16 π} (1 + \cos^{2} ϑ)$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin ϑ \cos ϑ$	$- \frac{3}{16 π} (1 - \cos^{2} ϑ)$	0
$E_{2,1, 0}^{*}$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ 2 i ϕ} \sin^{2} ϑ$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{1}{8 π} (1 + 3 \cos^{2} ϑ)$	$+ \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin ϑ \cos ϑ$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- 2 i ϕ} \sin^{2} ϑ$
$E_{2,1, - 1}^{*}$	0	$- \frac{3}{16 π} e^{+ 2 i ϕ} (1 - \cos^{2} ϑ)$	$+ \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{3}{16 π} (1 + \cos^{2} ϑ)$	$+ \frac{3}{8 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin^{2} ϑ$
$E_{2,1, - 2}^{*}$	0	0	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ 2 i ϕ} \sin^{2} ϑ$	$+ \frac{3}{8 π} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{3}{8 π} \sin^{2} ϑ$

	$E_{1,0, + 1}$	$E_{1,0, 0}$	$E_{1,0, - 1}$
$E_{2,1, + 2}^{*}$	$- \frac{1}{4 π} \sqrt{\frac{3}{2}} e^{+ i ϕ} \sin ϑ$	0	0
$E_{2,1, + 1}^{*}$	$+ \frac{1}{4 π} \sqrt{\frac{3}{2}} \cos ϑ$	$- \frac{\sqrt{3}}{8 π} e^{+ i ϕ} \sin ϑ$	0
$E_{2,1, 0}^{*}$	$- \frac{1}{8 π} e^{- i ϕ} \sin ϑ$	$+ \frac{1}{2 \sqrt{2} π} \cos ϑ$	$- \frac{1}{8 π} e^{+ i ϕ} \sin ϑ$
$E_{2,1, - 1}^{*}$	0	$- \frac{\sqrt{3}}{8 π} e^{- i ϕ} \sin ϑ$	$+ \frac{1}{4 π} \sqrt{\frac{3}{2}} \cos ϑ$
$E_{2,1, - 2}^{*}$	0	0	$+ \frac{1}{4 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin ϑ$

	$E_{1,0, + 1}$	$E_{1,0, 0}$	$E_{1,0, - 1}$
$E_{1,0, + 1}^{*}$	$- 1 / 4 π$	0	0
$E_{1,0, 0}^{*}$	0	$- 1 / 4 π$	0
$E_{1,0, - 1}^{*}$	0	0	$- 1 / 4 π$

	$E_{2,1, + 2}$	$E_{2,1, + 1}$	$E_{2,1, 0}$	$E_{2,1, - 1}$	$E_{2,1, - 2}$
$E_{2,1, + 2}^{*}$	$+ \frac{3}{8 π} \sin^{2} ϑ$	$- \frac{3}{8 π} e^{- i ϕ} \sin ϑ \cos ϑ$	$+ \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- 2 i ϕ} \sin^{2} ϑ$	0	0
$E_{2,1, + 1}$	$- \frac{3}{8 π} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{3}{16 π} (1 + \cos^{2} ϑ)$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin ϑ \cos ϑ$	$- \frac{3}{16 π} (1 - \cos^{2} ϑ)$	0
$E_{2,1, 0}^{*}$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ 2 i ϕ} \sin^{2} ϑ$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{1}{8 π} (1 + 3 \cos^{2} ϑ)$	$+ \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin ϑ \cos ϑ$	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{- 2 i ϕ} \sin^{2} ϑ$
$E_{2,1, - 1}^{*}$	0	$- \frac{3}{16 π} e^{+ 2 i ϕ} (1 - \cos^{2} ϑ)$	$+ \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{3}{16 π} (1 + \cos^{2} ϑ)$	$+ \frac{3}{8 π} \sqrt{\frac{3}{2}} e^{- i ϕ} \sin^{2} ϑ$
$E_{2,1, - 2}^{*}$	0	0	$- \frac{1}{8 π} \sqrt{\frac{3}{2}} e^{+ 2 i ϕ} \sin^{2} ϑ$	$+ \frac{3}{8 π} e^{+ i ϕ} \sin ϑ \cos ϑ$	$+ \frac{3}{8 π} \sin^{2} ϑ$

Interpreting angular momentum transfer between electromagnetic multipoles using vector spherical harmonics

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Abstract

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Tables (3)

Equations (11)

Optics Letters