Optical Resolution – the Diffraction Limit
Optical magnification instruments are meant to make tiny objects visible. For that reason, the objects are magnified to cover a sufficient area of our retina. Optical magnification is achieved by a single lens (magnifier) or lens systems (compound microscope). The compound microscope consists of an objective lens and an eyepiece. In modern microscopes, the “objective lens” may contain some 20 individual lenses – needed to correct for many kinds of optical aberrations. Beside the optical magnification, electronic magnification is also important. Here, a pixelated image is magnified e.g. on a computer screen, printer or beamer by increasing the pixel size.
Obviously, electronic magnification by changing pixel size does not alter the resolution. The information in the image is not changed; the structural features are only displayed on a different scale. Nevertheless, the apparent resolution may be improved by inflating the information until our eyes can recognize it.
Optical imaging (the step before electronic imaging) has an inherent capability to resolve structural details. This resolution power depends on two parameters only (for ideal optical systems): the wavelength and the solid angle the lens clips from a full sphere. In circular optical systems, this fraction is described by the numerical aperture (NA). The NA is a technical property of the objective lens and engraved in the lens barrel.
An infinitesimally small object (that is: a point) would be represented by an optical imaging system (e.g. a compound microscope) as an extended structure. In circular optical systems this structure is the Airy pattern, a pattern that is caused by the light waves diffracted at the fringes of the optical lens. (Usually, these fringes are introduced by “apertures”, i.e. diaphragms that confine the circular beam area.)
The faintest structure that is resolved by an optical imaging system is therefore in the range of the size of that diffraction pattern. The pattern may be described by the diameter of the inner disc (which is defined by the first zero of the intensity), by the full-width-half-maximum, by 1/e of the maximum intensity or other choices. These parameters are arbitrary and consequently the “resolution power” is also defined by convention. The most prominent convention was first given by Ernst Abbe in 1874 and reads:
Fig. 1: Left: Airy diffraction pattern created by a circular lens when illuminated homogeneously. The inner bright spot is called the "Airy disc" and is usually a measure of the size of the diffraction-limited spot image. The pattern is also referred to as focal "point spread function".
Right: Intensity profile across the center of the Airy pattern. The dark ring around the Airy disc is the position of the first zeroes on both sides of the peak. The distance of these zeroes (i.e. the diameter of the Airy disc) is 1.21*λ/NA.
Stimulated Emission – the Strange Interaction of Light and Matter
As pointed out, the limit of imaging structural details with optical systems is defined by the diffraction pattern that is created by that optical system from an infinitesimally tiny spot. So in order to see more details, other paths need to be found. The first path was discovered by Stefan Hell in 1994, employing the principle of stimulated emission. Stimulated emission is the least obvious interaction type of light and matter. It was first discussed by Albert Einstein in 1916.
Absorption: Once a photon hits a molecule (or an atom), the molecule’s electron system may be excited from a ground state to an excited state. The interaction is called resonant when the photon is absorbed and its energy is sponged up by the electron system – which is consequently “excited”. This absorption is one mode of interaction and it is the basis of fluorescence. Fluorescence, on the other hand, has evolved into the most important contrasting method in modern microscopy. This is where things start to join up.
Emission: A second type of interaction is the (spontaneous) release of a photon from an electron system of a molecule (or atom). In fluorescence, this emission process produces the photons that are collected to create images. The emission is not monochromatic, as there are a series of sub-states in both the excited and ground state, hence the energy of the photons is variable and the emission appears as a comparably broad spectrum – even if the excitation was monochromatic.
Stimulated Emission: When a photon (with appropriate photon energy) hits an excited electron system, the system can release its energy by emitting a photon - like a mouse trap that is loaded with a tennis ball might be tricked to release that ball by interaction with a tennis ball thrown on the trap. This type of interaction is the stimulated emission. In contrast to the tennis ball mousetrap example, the emitted photon always has the same properties as the incident photon. This effect is used to amplify light (like in the mousetrap example) and is the fundament of laser technology.
In STED, this effect is utilized to de-excite electron systems of fluorochromes in a desired area – like removing excitation with an eraser. The photons emitted in this process are not collected.
Playing with Diffraction – Torus-shaped Pattern
With the process of stimulated emission and the process of absorption we now have the tools to manipulate the excitation status of fluorochromes in a microscopic sample. The excitation pattern in a true confocal scanning microscope is identical with the Airy pattern as described above. If we apply a similar pattern for de-excitation, the residual excitation pattern would again represent an Airy pattern, the resolution would not be altered and ruled by the shortest wavelength applied. The tricky bit is to engineer the diffraction pattern of the beam that is meant for de-excitation in a way that reduces the area of excited fluorochromes. The inner part of the Airy pattern determines the resolution. In order to narrow that inner part, the de-excitation pattern must leave an inner part and erase the outer part. Consequently, the de-excitation pattern should be toroid.
How is a toroid diffraction pattern achievable? Fortunately, this task is comparably easy. If the illuminating light beam is partially phase-shifted, the interference pattern at the focal plane will show alterations. A toroid focus is achieved by a phase plate with a l/2 segment. Half the beam shifted by 90° creates a doublet. Two overlaying beams with rotated doublets make a sort of toroid. A helical phase-shift plate generates a perfect toroid. The job is done by simply inserting a single element in the de-excitation illumination part: an appropriate phase plate.
A STED microscope consequently is illuminating diffraction limited at an appropriate excitation wavelength. Coaxially with the same beam a second beam with appropriate de-excitation wavelength illuminates the sample with a toroid (donut) focus. The image is generated by scanning the sample as in a conventional scanning microscope.
Airy + Bagel + Einstein = Super-Resolution
The set-up described above creates an area of excited fluorochromes – the area is Airy-shaped (folded with the excitation properties of the fluorochrome). If the illumination is pulsed, the de-excitation (STED) pulse that follows shortly after will erase excitation of fluorochromes in a toroid around the center of the excitation focus. The STED wavelength is at the red edge of the emission band, and the fluorescence emission that originates from the residually excited molecules is collected between the two wavelengths.
Pulsed excitation and emission depletion is the best solution, but it is also possible to do the same job by cw illumination. Here, the focus is reduced by a dynamic equilibrium of excitation and de-excitation.
The residual area depends on the intensity of the de-excitation intensity. By increasing the STED intensity, an increasing fraction of the excited area will be de-excited. As long as the STED intensity in the center is zero, there will always be some residual area for emitting fluorescence photons. Consequently, the achievable resolution is potentially infinite. 70nm is a current standard. The record is below 10nm. This is some 20 times better than the conventional far-field optical resolution (wide field or confocal). In theory, there is no limit.