Fig. 1. (a) Schematic of one emitter with position (x0,y0,z0). (b) Schematic of two emitters with positions (x1,y1,z1) and (x2,y2,z2) and different intensities (q1,q2).

Fig. 2. Quantum and classical Fisher information of localization in 3D space. For estimation of the transverse coordinates of the emitter, the CFI coincides with the QFI in the position z=0, which indicates intensity measurement achieves QFI if the detector is put in the position of waist; for the estimation of the longitudinal coordinate, the detector needs to be put at the Rayleigh range to get the best precision.

Fig. 3. Contour plot of ℜ of two Gaussian incoherent beams model in three dimensions in the (t, s) plane. (a) Relative intensity is a constant and equals to 0.5. (b) Relative intensity is also a parameter to be estimated; here, we set q=0.1. (c) Similar to (b) while q=0.3. (d) Similar to (b) while q=0.5.

Table 1. Ratio between the QFI of Gaussian Beam and That of LG Beam with Respect to the Azimuthal Mode Index p and Radial Index l^a