. Author manuscript; available in PMC: 2020 Feb 27.

Published in final edited form as: J Opt Soc Am A Opt Image Sci Vis. 2019 Aug 1;36(8):1346–1360. doi: 10.1364/JOSAA.36.001346

Table 2.

Summary of relevant quantities for in-focus spatio-angular dipole imaging—see [1] for derivations. $F_{ℝ^{2}}$ denotes a two-dimensional Fourier transform, and $F_{S^{2}}$ denotes a spherical Fourier transform.

Quantity	Symbol	Relationships
Dipole emission density	$f (r_{o}^{⊥}, {\hat{s}}_{o})$	—
Dipole spatial spectrum	$F (ν, {\hat{s}}_{o})$	$= F_{ℝ^{2}} \{f (r_{o}^{⊥}, {\hat{s}}_{o})\}$
Dipole angular spectrum	$F_{l}^{m} (r_{o}^{⊥})$	$= F_{S^{2}} \{f (r_{o}^{⊥}, {\hat{s}}_{o})\}$
Dipole spatio-angular spectrum	$F_{l}^{m} (ν)$	$= F_{ℝ^{2}} \{F_{l}^{m} (r_{o}^{⊥})\} = F_{S^{2}} \{F (ν, {\hat{s}}_{o})\}$

Dipole coherent spread function	$c (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o})$	—
Dipole coherent transfer function	$C (τ, {\hat{s}}_{o})$	$= F_{ℝ^{2}} \{c (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o})\}$
Dipole point spread function	$h (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o})$	$= {\|c (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o})\|}^{2}$
Dipole spatial transfer function	$H (ν, {\hat{s}}_{o})$	$= F_{ℝ^{2}} \{h (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o})\}$ $= \int_{ℝ^{2}} d τ C (τ, {\hat{s}}_{o}) C^{†} (τ - ν, {\hat{s}}_{o})$
Dipole angular transfer function	$H_{l}^{m} (r_{d} - r_{o}^{⊥})$	$= F_{S^{2}} \{h (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o})\}$
Dipole spatio-angular transfer function	$H_{l}^{m} (ν)$	$= F_{ℝ^{2}} \{H_{l}^{m} (r_{d} - r_{o}^{⊥})\} = F_{S^{2}} \{H (ν, {\hat{s}}_{o})\}$

Scaled irradiance	g(r_d)	$= \int_{S^{2}} d {\hat{s}}_{o} \int_{ℝ^{2}} d r_{o}^{⊥} h (r_{d} - r_{o}^{⊥}, {\hat{s}}_{o}) f (r_{o}^{⊥}, {\hat{s}}_{o})$ $= \sum_{l m} \int_{ℝ^{2}} d r_{o}^{⊥} H_{l}^{m} (r_{d} - r_{o}^{⊥}) F_{l}^{m} (r_{o}^{⊥})$
Scaled irradiance spectrum	G(v)	$= F_{ℝ^{2}} \{g (r_{d})\}$ $= \int_{S^{2}} d {\hat{s}}_{o} H (ν, {\hat{s}}_{o}) F (ν, {\hat{s}}_{o})$ $= \sum_{l m} H_{l}^{m} (ν) F_{l}^{m} (ν)$