APPENDIX A
POINT SPREAD FUNCTION
Point Spread Function (PSF) is a spatial function of light intensity describing the output of
the imaging system (the microscope) for an input point source. As the most fundamental
characteristic of an imaging system, PSF is viewed as the impulse response function of a
microscope in spatial domain.106
For a shift-invariant, ideally corrected and aberrant free microscope, the PSF,
axisymmetric along the optical axis and symmetric about conjugate image plane, is given by,106
h(x,y,z)= C J, (vp)exp 2 dp (A-l)
In Equation A-1, (x,y, z) are coordinates from the conjugate point; J0 is the zeroth-order Bessel
function of the first kind; p, u, and v are dimensionless numbers defined as p =
a
2;r a2 2;r a
u = f + z and v = -- x + y where a is the radius of objective aperture, f is
A f(f + Z) f /
the focal length, and 2 is the wavelength of emission light; and C is a constant. The expression
given above is only one approximation for the three dimensional point spread function. Many
alternative forms of 3D PSF have been reported based on scalar diffraction theory.911' 61
The plot of PSF from Equation A-i is a 3D diffraction pattern, centered on the conjugate
image of the point source located in the image plane, as shown in Figure A-1.
On the image plane where z = 0, the PSF is simplified to h(x,y) = C J (vp)pdp ,
which represents a series of concentric spheres known as airy disk (Figure A-2). The radius of
airy disk (distance between the central maximum and the first minimum) is related to the
numerical aperture (NA) of the objective lens,
0.61A
'lateral- (A-2)
NA
In the axial direction, the intensity profile (Figure A-3) is similar to that on lateral
direction, and the distance between the maximum intensity of the central bright region to the first
point of minimum intensity along optical axis is given by
2An
r NA2 (A-3)
aalNA2