Chapter 4 – Making Light Work Harder in Biology  113

This indicates that the first-​order intensity minimum at an angle a satisfies

(4.2)

sin

m

α

λ

≈1 22 2

.

r

The Rayleigh criterion for optical resolution states that two PSF images can just be

resolved if the peak intensity distribution of one falls on the first-​order minimum of the other.

The factor of 1.22 comes from the circularity of the aperture; the wavelength λm in propa­

gating through an optical medium of refractive index n is λ/​n where λ is the wavelength in a

vacuum. Therefore, if f is the focal length of the objective lens, then the distance the optical

resolution in terms of displacement along the x axis in the focal plane Δx is

(4.3)

∆x

f

f

r

n

NA

m

=

=

=

=

sin

sin

max

θ

λ

λ

θ

λ

1 22

2

0 16

0 61

.

.

.

This is a modern version of the Abbe equation, with the optical resolution referred to as

the Abbe limit. Abbe defined this limit first in 1873 assuming an angular resolution of sin

θ =​ λm/​d for a standard rectangular diffraction grating of width d, with the later factor of

1.22 added to account for a circular aperture. An Airy ring diffraction pattern is a circularly

symmetrical intensity function with central peak containing ~84% of the intensity, such that

multiple outer rings contain the remaining intensity interspaced by zero minima.

The Rayleigh criterion, from the eponymous astronomer, based on observations of stars,

which appear to be so close together that they are difficult to resolve by optical imaging, is

FIGURE 4.1  Resolving fluorophores. (a) Airy disk intensity functions displayed as false-​

color heat maps corresponding to two identical fluorophores separated by 500 nm (clearly

resolved), 230 nm (just resolved), and 0 nm (not resolved—​in fact, located on top of each other).

(b) Methods to reduce the density of photoactive fluorophores inside a cell to ensure that the

concentration is less than Clim, at which the nearest-​neighbor photoactive fluorophore separ­

ation is equal to the optical resolution limit. (c) BiFC that uses genetic engineering technology

(see Chapter 7) to generate separate halves of a fluorescent protein that only becomes a single-​

photoactive fluorescence protein when the separate halves are within a few nanometers of

each other to permit binding via a leucine zipper.