SoSe 20 Exp. Quantum Optics Group / Laboratory of Nano-Optics
Experimental Methods in Nano and Quantum Optics
Exercise 2
The point spread function (PSF) characterizes the imaging capabilities of an optical system for a dipole source. There are two kinds of PSF. The transverse PSF describes the resolution power in the transverse direction of the image plane. The longitudinal PSF describes the resolution power in the axial direction. The calculation of the PSF can be complicated, but in the paraxial approximation analytical expressions can be obtained.
1. The paraxial PSF in the paraxial approximation reads,
, with .
represents the x-component of the electric filed in the image plane, px is a dipole source in the object plane, is the axial coordinate, is the numerical aperture and the transverse magnification of the optical system. In the following we assume
n=n’=1 and nm.
Fig. 1 Schematics for the derivation of the PSF (adapted from Wikipedia).
A. Plot the collection efficiency of the objective lens as a function of NA in the range from 0 to 1. The source is a px dipole at the focal point of the objective lens.
B. Repeat the previous task assuming a longitudinal dipole source pz.
C. Assuming M=1, plot the paraxial PSF for different values of NA in the range from 0 to 1.
D. Assuming NA=0.5, plot the paraxial PSF for different values of M in the range from 1 to 100.
E. Consider NA=0.5 and a photodetector with a circular area having diameter 25 m placed in the image plane. Calculate the detection efficiency as a function of M as the ratio between the total power carried by the paraxial PSF and the fraction of power that hits the photodetector.
F. Combining the result from A. and E., estimate the maximum efficiency as the product of the collection efficiency for NA=0.5 and the maximum detection efficiency.
θmax
lim
≪π/2| E(x, y,0) |
2= π
4ϵ
02nn′
p
x2λ
6NA
4M
2[ 2 J
1(2π ρ ˜ ) 2π ρ ˜ ]
2
ρ ˜ = NA ρ Mλ E(x, y,0)
ρ NA = n sin θ
maxM = (nf ′ )/(n′ f ) λ = 600
μ
n n’
f f’
θmax
px
2. Likewise, one can obtain the axial point spread function (PSF) in the paraxial approximation, which reads
, with .
represents the x-component of the electric filed in the axial direction.
A. Assuming M=1, plot the axial PSF for different values of NA in the range from 0 to 1.
B. Assuming NA=0.5, plot the axial PSF for different values of M in the range from 1 to 100.
C. For the case F. of the previous exercise, calculate how the efficiency depends on the axial position of the photodetector. For which axial displacement the efficiency has dropped by 50%?
D. Compare the result from C. to the so-called depth of field, given by the expression .
θmax
lim
≪π/2| E(0,0,z) |
2= π
4ϵ
02nn′
p
x2λ
6NA
4M
2[ sin(π z ˜ ) π z ˜ ]
2
