In microscopy, optical or spatial resolution is understood as the distance that two structures must have at least in order to be perceived as separate image structures after optical imaging. For example, the minimum distance between two point-shaped objects required for separate detection or the minimum distance between lines in an optical grid is considered.
The achievable resolution in classical light microscopy is fundamentally limited by the fact that the optical near fields surrounding the object are not transmitted through the optical system, which is sometimes also referred to as diffraction in free space. This minimum distance from the object is called the resolution limit or Abbe limit . Ernst Abbe described this relationship in the 19th century. More recent methodological approaches allow a resolution well beyond this limit; they are collectively referred to as English Superresolution Microscopy ( German super-resolution microscopy ). Such techniques are, for example, RESOLFT microscopy with STED microscopy , microscopy with modulated illumination (SIM) , photoactivated localization microscopy (STORM microscopy) and optical scanning near- field microscope .
There are different approaches to determine the achievable resolution. Abbe started from a grid with closely spaced lines through which light shone and calculated how dense the lines can be so that they can just about be resolved as lines. Abbe examined illuminated (passive) objects. John William Strutt, 3rd Baron Rayleigh, looked at point sources of light. He described the distance between active objects that could just be recognized as separate. Related to this approach is the determination of the half width of the microscopic signal of a point light source. All three approaches mathematically lead to very similar values for the resolution.
To enable the theoretically possible resolution, it is necessary that enough light be collected. This is usually not a problem with bright field microscopy . In fluorescence microscopy , low intensities in connection with short exposure times can lead to too few photons reaching the detector and the contrast achieved being insufficient for the separate recognition of the structures.
A distinction must be made between the resolution and the detectability and the achievable positioning accuracy. With dark field microscopy , especially ultramicroscopy , or fluorescence microscopy, it is still possible to detect particles whose size is considerably below the limit of resolution, with fluorescence microscopy down to individual dye molecules. Positioning accuracy is about determining the position of a surface or a body as precisely as possible in space. For this purpose, the maximum brightness of the light emanating from a body can be determined. This is possible with an accuracy in the nanometer range. In both cases, however, the resolution does not fall below the resolution limit: For example, it is not possible to determine whether the emitted light actually originates from a point-like structure or from several that are close together.
Derivation and scope
Ernst Abbe examined the diffraction behavior of light on line gratings . The closer the lines of the grid are to each other (i.e. the more lines there are in a millimeter), the more light that passes through is diffracted. First, the case is considered in which the light strikes (only) perpendicularly on the back of the grille (central lighting). In order to obtain information about the grating structure to be imaged, in addition to the main maximum of the diffraction pattern passing straight (perpendicular) through the grating, at least the first secondary maximum must also be picked up. The opening (more precisely: the opening angle ) of the lens must be sufficiently large for this. If the lines of the grating are so close together that the first secondary maximum can no longer be picked up by the objective, the grating structure is no longer resolved.
Abbe wrote about these connections in 1873 in his work "Contributions to the theory of the microscope and microscopic perception":
"[...] the physical differentiation limit [...] depends solely on the opening angle and is proportional to the sine of half its amount."
and more precisely elsewhere:
"Since the opening angle in the immersion system cannot by any means be extended significantly beyond the size that would correspond to 180 ° in air, it follows that [...] the differentiation limit for central illumination is never over the amount of the whole, and for extreme oblique lighting will never go beyond half the wavelength of blue light by a notable amount. "
“Inclined lighting” here means lighting with a condenser , which causes the light to strike the specimen as a cone of rays and not as a cylinder as with central lighting. Correct lighting, which also includes light falling at an angle, is essential to achieve the maximum possible resolution. In addition, the importance of the wavelength becomes clear: Since short-wave (blue) light is less diffracted than long-wave (red) light, better resolution can be achieved with short-wave light. The Abbe criterion is absolute; it cannot be overcome with conventional microscopy.
Abbe's considerations on image formation apply to cases in which incident light is diffracted by the specimen and then passed on to the detector. Since the lighting comes from a light source, it can be considered coherent . Abbe's considerations do not make any statements for self-luminous objects. No information is given for the resolution along the optical axis (z-direction) either.
The following applies to all formulas: is the achievable resolution, more precisely: the distance that two lines in a grid must have at least so that they can still be recognized as separate lines in the microscope.
In the case of exclusively central lighting without a condenser, this results
or, since the numerical aperture (NA) is,
Objective and condenser with the same NA
The “extreme oblique illumination” mentioned by Abbe describes a transmitted light beam path in which a condenser is used for illumination that has at least the same aperture angle or at least the same numerical aperture as the objective. In this case, Abbe's formula results in its most well-known form:
or, because of shortened
Condenser with a smaller NA than objective
If the numerical aperture of the condenser and objective are the same size, a practical problem arises: the image contrast is very low. An optimal image contrast is achieved when the numerical aperture of the condenser is 2/3 of the numerical aperture of the objective. If necessary, the condenser diaphragm (= aperture diaphragm) can be closed to achieve this. Under the condition that the condenser aperture is smaller than the objective aperture, the following formula applies:
This version can also be viewed as a general form, since it applies with central illumination and with apertures of the same size again 2NA results below the fraction line.
- Objective with NA 0.25 (with many 10x objectives); Condenser with NA 0.55, yellow-green light with 550 nm, as the eye is most sensitive to this color range.
The condenser aperture can have the same value as the objective aperture, so in the case of a larger condenser aperture the objective aperture is taken twice.
- Objective with NA 0.65 (with many 40x objectives); Condenser with 0.55, light with 550 nm.
- Oil immersion objective with NA 1.4 and oil immersion condenser with 1.4, light with 550 nm
Derivation and scope
Separate principles apply to the resolution of self-luminous structures, such as those that occur in fluorescence microscopy. The first description of these rules for microscopy is, depending on the textbook , traced back either to a work from 1896 by John William Strutt, 3rd Baron Rayleigh , or to Rayleigh and Hermann von Helmholtz, or only to work by Helmholtz from 1873 or 1874. In all cases the corresponding criterion is called the Rayleigh criterion , which was transferred from astronomy or from the telescope to the microscope.
Microscopic images of self-luminous points are not points, but diffraction patterns whose brightness distribution depends on the point spread function . The plane of the pattern which contains the highest intensity (sharpness plane) is called the diffraction disk. The Rayleigh criterion states that two self-luminous points that lie next to each other in the focal plane can just be distinguished when the intensity maximum of the diffraction disk of the first point falls into the minimum of the diffraction disk of the second point (see figure). The brightness of the darkest point between the two maxima is then 73.5% of the maxima. The Rayleigh criterion is therefore a convention and not an absolute criterion, since some observers may also be able to distinguish diffraction disks that are even closer together.
The formulas listed in this section apply to self-illuminating points when observing with classical microscopy, for example in classical fluorescence microscopy . For confocal microscopy and multiphoton excitation see the relevant sections.
The Rayleigh criterion also includes the wavelength, but not the wavelength of the light diffracted at the object, as is the case with the Abbe limit, but the wavelength of the light emitted by the object, i.e. the wavelength of the fluorescence in fluorescence microscopy.
The resolution (Rayleigh criterion) in the focal plane is
- , Sometimes given in the form ,
where NA is the numerical aperture of the objective used and λ is the wavelength of the emitted light.
The resolution of a microscope is generally poorer along the optical axis (z-direction) than within the focal plane. For self-luminous points (classic fluorescence microscopy), as with the Rayleigh criterion, the distance between the brightness maximum of the diffraction pattern and the first minimum along the optical axis can also be calculated along the optical axis. Due to the underlying theoretical assumptions, the resulting formula only applies to paraxial optics , i.e. for objectives with small opening angles and a low numerical aperture.
where the refractive index of the medium between the objective and the cover slip or preparation is, for example, 1 for air for dry objectives or 1.518 for typical immersion oil . A determination of the intensity distribution for objectives with a higher numerical aperture can be achieved using the Fresnel-Huygens theory , with fewer assumptions being required and thus a more realistic distribution resulting.
While the lateral resolution is always the same for objectives with the same numerical aperture , this formula shows that immersion objectives in the axial direction with the same numerical aperture have a poorer resolution, since the refractive index n of the immersion medium is also entered here above the fraction line. However, since immersion objectives usually have a higher numerical aperture than dry objectives, this aspect rarely comes into play, since a higher square numerical aperture contributes to an improvement in the resolution.
Derivation and scope
The resolution according to the Rayleigh criterion can be calculated, but it is difficult to determine experimentally: Two very small objects would have to be pushed closer and closer together until they could no longer be distinguished. For practical reasons, therefore makes do in fluorescence microscopy with the half-width (ger .: full width half maximum of, FWHM) point spread function (Engl. Point spread function , PSF). The point spread function describes the three-dimensional image of a fluorescent point created by diffraction; it is a function of the optical system, essentially the objective. It can also be calculated. For the experimental determination, fluorescent objects are used, the size of which is below the resolution limit, for example 150 nanometers small latex spheres or quantum dots soaked with fluorescent dye . The half-width of the intensity curve is measured in nanometers or micrometers (see figure). The resolution and half width of the PSF are mathematically related; the PSF half width is slightly smaller than the resolution.
The formulas listed in this section apply to self-illuminating points when observing with classical microscopy, for example in classical fluorescence microscopy. For confocal microscopy and multiphoton excitation see the relevant sections.
In the x, y-direction the half width of the PSF is included
where NA is the numerical aperture of the objective used and λ is the wavelength of the emitted light.
Along the optical axis (z-direction) the half width of the PSF is included
This formula can be shortened to for numerical apertures below 0.5
For an oil immersion objective with a numerical aperture of 1.4 and a wavelength of 550 nm, the result in the focal plane (x, y) is a half width of
Along the optical axis when using an immersion oil with the refractive index n = 1.518 results
Increasing the resolution in classical microscopy
Both the Abbe limit and the Rayleigh criterion show that the resolution can be increased down to smaller values in two ways: by reducing the wavelength λ and by increasing the numerical aperture NA.
Shortening the wavelength
The approach of reducing the wavelength led to the development of UV microscopy at the beginning of the 20th century , in which UV light was used instead of visible light. August Köhler developed corresponding devices from 1900 at Zeiss. This enabled the resolution to be doubled at a wavelength that was halved compared to visible light. Since the human eye cannot perceive this light, it can only be displayed using film or fluorescent screens . In addition, lenses, microscope slides and so on must be made of quartz glass , fluorspar or lithium fluoride , since normal glass absorbs UV light. Due to these difficulties, UV microscopy could not generally prevail and was reserved for special applications.
Increase in the numerical aperture
The numerical aperture NA is the product of n, the refractive index of the medium between the specimen and the objective, and sinα, the sine of half the opening angle. In order to increase NA, one of the two factors must be increased. Assuming an objective would have an infinitely large front lens, the opening angle would be 180 °, half the opening angle would be 90 ° and its sine would be 1.Since the front lens of the lens is finite and a certain working distance between the focal plane and the front lens is required sinα always less than 1. High quality lenses can achieve values of 0.95.
The refractive index can be increased by using immersion , i.e. a liquid is introduced between the specimen and the objective. For this purpose, special objectives are necessary for different immersion media, the lenses of which have been calculated in such a way that they are adapted to the changed refraction behavior in front of the objective. While air has a refractive index of about 1, that of water is 1.33 and that of typical immersion oil is 1.518. Correspondingly, the resolution with oil immersion can be increased by over 50% compared to immersion-free microscopy.
The highest numerical apertures with wider distribution are therefore NA = 1.4 for oil immersion objectives with a magnification of 100 × or 60 ×. Oil immersion condensers can also achieve NA = 1.4. Monobromonaphthalene , which Ernst Abbe introduced to microscopy, has an even higher refractive index than oil with n = 1.666. Corresponding objectives achieved an NA of 1.63. However, they were not suitable for the transmitted light microscopy of biological objects, since microscope slides, cover slips, embedding medium and the actual object would have to have at least the same refractive index for their use.
Numerical apertures higher than 1.4 are found today with 1.45 in special oil immersion objectives for TIRF microscopy.
Confocal fluorescence microscopy
In conventional fluorescence microscopy, the sample is illuminated over a large area and imaged through the objective so that the PSF of the overall system is only determined by the (detection) PSF of the objective. In contrast to this, the sample is illuminated in the confocal microscope with the diffraction disk (more precisely: the point spreading function) of the illumination pinhole. In this case, the resolution of the overall system is determined by multiplying two point spread functions, namely the illumination PSF and the detection PSF.
Resolution in the focal plane (x, y)
The PSF multiplication does not change the diameter of the diffraction disk, the first minimum is still in the same place as before. However, the increase in brightness is now steeper, the edges of the brightness distribution move inwards. Two such multiplied diffraction disks are therefore closer to each other when the minimum intensity between the maxima reaches the 73.5% mentioned above for the Rayleigh criterion; one then speaks of the generalized Rayleigh criterion. Theoretically, this results in an improvement in resolution by the factor , i.e. about .
In the focal plane this would result in the half-width of the signal of a fluorescent point-shaped object or , with the mean value of the excitation and emission wavelengths being used as the wavelength for fluorescence. With green light (wavelength 500 nm) this corresponds to a half width of 132 nm (non-confocal: 182 nm) for an oil immersion objective with 1.4 numerical aperture. However, this value can only theoretically be achieved with an infinitely small pinhole, which means that no more light would be captured. It is also assumed that the stimulating light completely fills the lens from the rear. However, this is not always the case.
The improvement that can be achieved in practice is therefore less. It depends on how much of the diffraction disk is let through by the pinhole opening, i.e. on the diameter of the pinhole. If the aperture diameter is in the first minimum of the diffraction disk, the resolution in the focal plane is no longer better than in the non-confocal case, whereas the signal intensity is then almost at its maximum. However, excluding fluorescence from other levels still works quite well. Therefore, this value is often preset in commercial confocal microscopes. It is called an Airy unit (AU), after the English terms Airy disk (= diffraction disk ) and unit (= unit of measurement).
Resolution along the optical axis
Here too, as in the focal plane, there is theoretically an improvement by the factor , i.e. for the half-width of the PSF of a fluorescent point object
The wavelength is in turn the mean value of the excitation light and fluorescent light.
A thin fluorescent layer has a similar but slightly different equation for the half width:
The same practical restrictions result for this theoretical improvement in resolution that were already described in the previous section for the improvement in resolution in the focus plane.
Excitation with two photons occurs in two-photon fluorescence microscopy and in second harmonic generation microscopy, see multiphoton microscope . Similar to confocal microscopy , the resolution is therefore determined by multiplying two point spread functions (PSF), except that it is the excitation PSF twice. So there is again an improvement by the factor . Since, in contrast to confocal microscopy, no pinhole diaphragm is used with multiphoton excitation, the corresponding restriction does not apply and the improved resolution is actually possible here. The size of the PSF depends exclusively on the excitation wavelength. However, since excitation wavelengths of 800 nm or more are generally used in these techniques, the resolution measured in micrometers does not result in a better resolution than in classical fluorescence or confocal microscopy. As with confocal microscopy, for maximum resolution it is also necessary here that the stimulating laser beam completely fills the objective from the rear.
For the half width of the PSF in the focal plane (x, y) results
and thereby, for example, for two-photon fluorescence excitation with a water immersion objective with NA = 1.1
whereas normal, one-photon fluorescence excitation for the same dye with the same objective could result, for example, from the formulas listed above
The formulas given above for a confocal microscope with a closed pinhole diaphragm can also be used along the optical axis (z-direction).
With an excitation with three photons, i.e. with three-photon fluorescence microscopy and with third-harmonic-generation microscopy , there is an improvement by the factor . With the same excitation wavelength, a better resolution results for three-photon excitation than for two-photon excitation. The formula for the half width of the PSF in the focal plane (x, y) is
The theoretically possible resolution according to the formulas given above can only be achieved if the contrast between the light and dark objects is high enough so that they can actually be differentiated. This can become a problem, for example, with bright field microscopy of unstained biological specimens if they hardly absorb light. In many cases, the contrast can be improved if contrast-increasing methods such as dark field microscopy , phase contrast or differential interference contrast are used.
Especially in fluorescence microscopy applications, the problem can arise that the fluorescence generated is very weak and therefore a certain exposure time must not be exceeded in order to collect enough photons to achieve the required minimum contrast or a sufficient signal-to-noise ratio . In brightfield microscopy, this is generally not a problem due to the relatively high light intensities used; however, the phenomenon can be artificially simulated by greatly reducing the exposure level (see illustration).
Digitizing the maximum resolution: the Nyquist criterion
When microscopic images are digitized, the optically achieved resolution can only be obtained if the image points ( pixels ) are close enough together in the digital image . If one imagines two separate bright points lying next to one another, then these can only be displayed separately in the digital image if, to put it simply, there is a darker pixel between them: there would therefore be three pixels at the minimum representable distance, two for the bright ones Points and a dark one in between. In fact, the Nyquist criterion states that 2.3 pixels are required for the smallest optically representable distance. If, for example, the optically achievable resolution is 230 nm, the size of the pixels must therefore be 100 nm. A slightly more precise scan, i.e. slightly smaller pixels, can be justified in practice in order to make small differences more visible. However, scanning with more than 3 pixels ( oversampling ) does not bring any further advantages. The image file would be unnecessarily large. In the case of fluorescence microscopy applications, there is also the fact that the fluorescent dyes bleach unnecessarily quickly due to the accompanying stronger illumination . Other microscopic preparations that are sensitive to strong lighting can also be unnecessarily damaged.
The Nyquist criterion applies both to the focal plane (x, y) and along the optical axis (z-direction). The type of digitization is irrelevant, for example as parallel raster detection using a CCD / CMOS camera or with the aid of a fast photodetector and a scanner in the case of rastering processes.
Experimental verification of the achievable resolution
Particularly with super-resolution techniques (e.g. STED or STORM), the actually achievable resolution cannot always be easily derived from theory. However, there are a few ways to determine this experimentally. The currently most widespread method for this uses structures of the eukaryotic cytoskeleton , as these can be used to achieve very impressive images. The image obtained is searched for closely adjacent filaments that run as parallel as possible and their distance from one another is determined. The smallest value found is counted as the achieved resolution.
Another possibility is based on the use of nanometer rulers . These are DNA-based nanostructures that carry dye molecules at precisely defined distances from one another. They thus enable a less arbitrary and clearly more systematic determination of the optical resolution.
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