114 4.2  Super-Resolution Microscopy

semiarbitrary in suggesting that two objects could be resolved using circular optics if the

central maximum of the Airy ring pattern of one object is in the same position as the first-​

order minimum of the other, implying the Abbe limit of ~0.61λ/​NA. However, there are other

suggested resolution criteria that are perfectly reasonable, in that whether or not objects

are resolved is as much to do with neural processing of this intensity information in the

observer’s brain as it is with physics. One such alternative criterion calls for there being no

local dip in intensity between the neighboring Airy disks of two close objects, leading to the

Sparrow limit of ~0.47λ/​NA. The experimental limit was first estimated quantitatively by the

astronomer Dawes, concluding that two similarly bright stars could be just resolved if the dip

in intensity between their images was no less than 5%, which is in fact closer to the Sparrow

limit than the Rayleigh criterion by ~20%. Note that there is in fact a hard upper limit to

optical resolution beyond which no spatial frequencies are propagated (discussed later in this

chapter).

In other words, the optical resolution is roughly identical to the PSF width in a vacuum.

For visible light emission, the optical resolution limit is thus in the range ~250–​300 nm

for a high-​magnification microscope system, two orders of magnitude larger than the

length scale of a typical biomolecule. The z resolution is worse still; the PSF parallel to

the optic axis is stretched out by a factor of ~2.5 for high-​magnification objective lenses

compared to its width in x and y (see Chapter 3). Thus, the optical resolution limits in x,

y, and z are at least several hundred nanometers, which could present a problem since

many biological processes are characterized by the nanoscale length scale of interacting

biomolecules.

4.2.2  LOCALIZATION MICROSCOPY: THE BASICS

The most straightforward super-​resolution techniques are localization microscopy methods.

These involve mathematical fitting of the theoretical PSF, or a sensible approximation to it,

to the experimentally measured diffraction-​limited intensity obtained from a pixel array

detector on a high-​sensitivity camera. The intensity centroid (see Chapter 8 for details of the

computational algorithms used) is generally the best estimate for the location of the point

source emitter (the method is analogous to pinpointing with reasonable confidence where

the peak of a mountain is even though the mountain itself is very large). In doing so, the

center of the light intensity distribution can be estimated to a very high spatial precision, σx,

which is superior to the optical resolution limit. This is often performed using a Gaussian

approximation to the analytical PSF (see Thompson et al., 2002):

(4.4)

σ

π

x

s

a

N

s b

aN

=

+

+

2

2

2

2

2

12

4

/

where

s is the width of the experimentally measured PSF when approximated by a Gaussian

function

a is the edge length of a single pixel on the camera detector multiplied by the magnifica­

tion between the sample and the image on the camera

b is the camera detector dark noise (i.e., noise due to the intensity readout process and/​

or to thermal noise from electrons on the pixel sensor)

N is the number of photons sampled from the PSF

The distance w between the peak of the Airy ring pattern and a first-​order minimum is related

to the Gaussian width s by s ≈ 0.34w.

The Gaussian approximation has merits of computational efficiency compared to using

the real Airy ring analytical formation for a PSF, but even so is still an excellent model for

2D PSF images in widefield imaging. The aforementioned Thompson approximation results

in marginally overoptimistic estimates for localization precision, and interested readers are