Diffraction-limited system

The Abbe diffraction limit for a microscope

The observation of sub-wavelength structures with microscopes is difficult because of the Abbe diffraction limit. Ernst Abbe first mentioned the diffraction limit in his 1873 paper, page 466: „[…] die physikalische Unterscheidungsgrenze […] hängt allein vom Oeffnungswinkel ab und ist dem Sinus seines halben Betrages proportional“, or "[…] the physical limit of resolution […] depends solely on the aperture angle and is proportional to the sine of half its magnitude".^[2] Abbe wrote it in the form of a formula in his 1882 paper, page 461: "The smallest dimensions which are within the reach of a given aperture are indicated with sufficient accuracy by taking the limit of the resolving or separating power of that aperture for periodic or regular structures, i.e. the minimum distance apart at which given elements can be delineated separately with the aperture in question. The numerical expression of that minimum distance is" ^[3]

${\displaystyle d={\frac {\lambda }{2n\sin \theta }}={\frac {\lambda }{2\mathrm {NA} }}}$,

where ${\displaystyle \lambda }$ is the wavelength, ${\displaystyle n}$ is the refractive index of the medium, and ${\displaystyle \theta }$ is the semi-angle of the light focused by the optical system. The same formula had been proven by Hermann von Helmholtz in 1874.^[4] The portion of the denominator ${\displaystyle n\sin \theta }$ is called the numerical aperture (NA) and can reach about 1.4–1.6 in modern optics, hence the Abbe limit is ${\displaystyle d={\frac {\lambda }{2.8}}}$.

Considering green light around 500 nm and a NA of 1, the Abbe limit is roughly ${\displaystyle d={\frac {\lambda }{2}}=250{\text{ nm}}}$ (0.25 μm), which is small compared to most biological cells (1 μm to 100 μm), but large compared to viruses (100 nm), proteins (10 nm) and less complex molecules (1 nm). To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes. These techniques offer better resolution but are expensive, suffer from lack of contrast in biological samples and may damage the sample.

Digital photography

In a digital camera, diffraction effects interact with the effects of the regular pixel grid. The combined effect of the different parts of an optical system is determined by the convolution of the point spread functions (PSF). The point spread function of a diffraction limited circular-aperture lens is simply the Airy disk. The point spread function of the camera, otherwise called the instrument response function (IRF) can be approximated by a rectangle function, with a width equivalent to the pixel pitch. A more complete derivation of the modulation transfer function (derived from the PSF) of image sensors is given by Fliegel.^[5] Whatever the exact instrument response function, it is largely independent of the f-number of the lens. Thus at different f-numbers a camera may operate in three different regimes, as follows:

1. In the case where the spread of the IRF is small with respect to the spread of the diffraction PSF, in which case the system may be said to be essentially diffraction limited (so long as the lens itself is diffraction limited).
2. In the case where the spread of the diffraction PSF is small with respect to the IRF, in which case the system is instrument limited.
3. In the case where the spread of the PSF and IRF are similar, in which case both impact the available resolution of the system.

The spread of the diffraction-limited PSF is approximated by the diameter of the first null of the Airy disk,

${\displaystyle d/2=1.22\lambda N,\,}$^[6]

where ${\displaystyle \lambda }$ is the wavelength of the light and ${\displaystyle N}$ is the f-number of the imaging optics, i.e., ${\displaystyle 2\mathrm {NA} \rightarrow (2.44N)^{-1}}$ in the Abbe diffraction limit formula. For instance, for an f/8 lens (${\displaystyle N=8}$ and ${\displaystyle \mathrm {NA} \approx 2.5\%}$ ) and for green light (${\displaystyle \lambda _{g}=}$ 0.5 μm wavelength) light, the focusing spot diameter will be d = 9.76 μm or 19.5${\displaystyle \lambda _{g}}$. This is similar to the pixel size for the majority of commercially available 'full frame' (43mm sensor diagonal) cameras and so these will operate in regime 3 for f-numbers around 8 (few lenses are close to diffraction limited at f-numbers smaller than 8).

Cameras with smaller sensors will tend to have smaller pixels, but their lenses will be designed for use at smaller f-numbers and it is likely that they will also operate in regime 3 for those f-numbers for which their lenses are diffraction limited. Given the same field of view, pixel count, shutter speed and shot noise SNR (i.e. the same amount of light collected per pixel), a small sensor and a large sensor of equivalent quality will produce the same digital image, with the same amount of blur due to both diffraction and depth of field.^[7] The larger sensors do have an advantage in that the lenses for them tend to have larger maximum entrance pupils, which allows, to a larger extent, trading depth of field for less diffraction blur and either more SNR or higher shutter speeds.^[8]

Extending numerical aperture

The effective resolution of a microscope can be improved by illuminating from the side.

In conventional microscopes such as bright-field or differential interference contrast, this is achieved by using a condenser. Under spatially incoherent conditions, the image is understood as a composite of images illuminated from each point on the condenser, each of which covers a different portion of the object's spatial frequencies.^[12] This effectively improves the resolution by, at most, a factor of two.

Simultaneously illuminating from all angles (fully open condenser) drives down interferometric contrast. In conventional microscopes, the maximum resolution (fully open condenser, at N = 1) is rarely used. Further, under partially coherent conditions, the recorded image is often non-linear with object's scattering potential—especially when looking at non-self-luminous (non-fluorescent) objects.^[13] To boost contrast, and sometimes to linearize the system, unconventional microscopes (with structured illumination) synthesize the condenser illumination by acquiring a sequence of images with known illumination parameters. Typically, these images are composited to form a single image with data covering a larger portion of the object's spatial frequencies when compared to using a fully closed condenser (which is also rarely used).

Another technique, 4Pi microscopy, uses two opposing objectives to double the effective numerical aperture, effectively halving the diffraction limit, by collecting the forward and backward scattered light. When imaging a transparent sample, with a combination of incoherent or structured illumination, as well as collecting both forward, and backward scattered light it is possible to image the complete scattering sphere.

Unlike methods relying on localization, such systems are still limited by the diffraction limit of the illumination (condenser) and collection optics (objective), although in practice they can provide substantial resolution improvements compared to conventional methods.

Near-field techniques

The diffraction limit is only valid in the far field as it assumes that no evanescent fields reach the detector. Various near-field techniques that operate less than ≈1 wavelength of light away from the image plane can obtain substantially higher resolution. These techniques exploit the fact that the evanescent field contains information beyond the diffraction limit which can be used to construct very high resolution images, in principle beating the diffraction limit by a factor proportional to how well a specific imaging system can detect the near-field signal. For scattered light imaging, instruments such as near-field scanning optical microscopes and nano-FTIR, which are built atop atomic force microscope systems, can be used to achieve up to 10-50 nm resolution. The data recorded by such instruments often requires substantial processing, essentially solving an optical inverse problem for each image.

Metamaterial-based superlenses can image with a resolution better than the diffraction limit by locating the objective lens extremely close (typically hundreds of nanometers) to the object.

In fluorescence microscopy the excitation and emission are typically on different wavelengths. In total internal reflection fluorescence microscopy a thin portion of the sample located immediately on the cover glass is excited with an evanescent field, and recorded with a conventional diffraction-limited objective, improving the axial resolution.

However, because these techniques cannot image beyond 1 wavelength, they cannot be used to image into objects thicker than 1 wavelength which limits their applicability.

Far-field techniques

Far-field imaging techniques are most desirable for imaging objects that are large compared to the illumination wavelength but that contain fine structure. This includes nearly all biological applications in which cells span multiple wavelengths but contain structure down to molecular scales. In recent years several techniques have shown that sub-diffraction limited imaging is possible over macroscopic distances. These techniques usually exploit optical nonlinearity in a material's reflected light to generate resolution beyond the diffraction limit.

Among these techniques, the STED microscope has been one of the most successful. In STED, multiple laser beams are used to first excite, and then quench fluorescent dyes. The nonlinear response to illumination caused by the quenching process in which adding more light causes the image to become less bright generates sub-diffraction limited information about the location of dye molecules, allowing resolution far beyond the diffraction limit provided high illumination intensities are used.