Diffraction-limited system

Memorial to Ernst Karl Abbe, who approximated the diffraction limit of a microscope as

d=\frac{\lambda}{2n\sin{\theta}}

, where d is the resolvable feature size, λ is the wavelength of light, n is the index of refraction of the medium being imaged in, and θ (depicted as α in the inscription) is the half-angle subtended by the optical objective lens.

Log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.

The resolution of an optical imaging system – a microscope, telescope, or camera – can be limited by factors such as imperfections in the lenses or misalignment. However, there is a fundamental maximum to the resolution of any optical system which is due to diffraction. An optical system with the ability to produce images with angular resolution as good as the instrument's theoretical limit is said to be diffraction limited.^[1]

The resolution of a given instrument is proportional to the size of its objective, and inversely proportional to the wavelength of the light being observed. For telescopes with circular apertures, the size of the smallest feature in an image that is diffraction limited is the size of the Airy disc. As one decreases the size of the aperture in a lens, diffraction increases. At small apertures, such as f/22, most modern lenses are limited only by diffraction.

In astronomy, a diffraction-limited observation is one that is limited only by the optical power of the instrument used. However, most observations from Earth are seeing-limited due to atmospheric effects. Optical telescopes on the Earth work at a much lower resolution than the diffraction limit because of the distortion introduced by the passage of light through several kilometres of turbulent atmosphere. Some advanced observatories have recently started using adaptive optics technology, resulting in greater image resolution for faint targets, but it is still difficult to reach the diffraction limit using adaptive optics.

Radiotelescopes are frequently diffraction-limited, because the wavelengths they use (from millimeters to meters) are so long that the atmospheric distortion is negligible. Space-based telescopes (such as Hubble, or a number of non-optical telescopes) always work at their diffraction limit, if their design is free of optical aberration.

The beam from a laser with near-ideal beam propagation properties may be described as being diffraction-limited. A diffraction-limited laser beam, passed through diffraction-limited optics, will remain diffraction-limited, and will have a spatial or angular extent essentially equal to the resolution of the optics at the wavelength of the laser.

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 found in 1873 that light with wavelength λ, traveling in a medium with refractive index n and converging to a spot with angle $\theta$ will make a spot with radius

d=\frac{ \lambda}{2 n \sin \theta}

^[2]

The denominator $n\sin \theta$ is called the numerical aperture (NA) and can reach about 1.4–1.6 in modern optics, hence the Abbe limit is d = λ/2.8. Considering green light around 500 nm and a NA of 1, the Abbe limit is roughly d = λ/2 = 250 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.

Implications for 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 lens is simply the Airy disc. 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.^[3] Whatever the exact instrument response function we may note that 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:

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).
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.
in the case where the spread of the PSF and IRF are of the same order of magnitude, 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,

d/2 = 1.22 \lambda N,\,

where λ is the wavelength of the light and N is the f-number of the imaging optics. For f/8 and green (0.5 μm wavelength) light, d = 9.76 μm. This is of the same order of magnitude as 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.

Obtaining higher resolution

There are techniques for producing images that appear to have higher resolution than allowed by simple use of diffraction-limited optics.^[4] Although these techniques improve some aspect of resolution, they generally come at an enormous increase in cost and complexity. Usually the technique is only appropriate for a small subset of imaging problems, with several general approaches outlined below.

Extending numerical aperture

For a given numerical aperture (NA), the resolution of microscopy for flat objects under coherent illumination can be improved using interferometric microscopy. Using the partial images from a holographic recording of the distribution of the complex optical field, the large aperture image can be reconstructed numerically.^[5] Another technique, 4 Pi microscopy uses two opposing objectives to double the effective numerical aperture, effectively halving the diffraction limit.

Among sub-diffraction limited techniques, structured illumination holds the distinction of being one of the only methods that can work with simple reflectance without the need for special dyes or fluorescence and at very long working distances. In this method, multiple spatially modulated illumination patterns are used to double the effective numerical aperture. In principle, the technique can be used at any range and on any target provided that illumination can be controlled. Additionally, if exogenous contrast agents are used, the technique can also achieve more than a two-fold increase in resolution.

Near-field techniques

The diffraction limit is only valid in the far field. 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 far into the near field an imaging system extends. Techniques such as total internal reflectance microscopy and metamaterials-based superlens can image with resolution better than the diffraction limit by locating the objective lens extremely close (typically hundreds of nanometers) to the object. 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.

Laser beams

The limits on focusing or collimating a laser beam are very similar to the limits on imaging with a microscope or telescope. The only difference is that laser beams are typically soft-edged beams. This non-uniformity in light distribution leads to a coefficient slightly different from the 1.22 value familiar in imaging. But the scaling is exactly the same.

The beam quality of a laser beam is characterized by how well its propagation matches an ideal Gaussian beam at the same wavelength. The beam quality factor M squared (M²) is found by measuring the size of the beam at its waist, and its divergence far from the waist, and taking the product of the two, known as the beam parameter product. The ratio of this measured beam parameter product to that of the ideal is defined as M², so that M²=1 describes an ideal beam. The M² value of a beam is conserved when it is transformed by diffraction-limited optics.

The outputs of many low and moderately powered lasers have M² values of 1.2 or less, and are essentially diffraction-limited.

Other waves

The same equations apply to other wave-based sensors, such as radar and the human ear.

As opposed to light waves (i.e., photons), massive particles have a different relationship between their quantum mechanical wavelength and their energy. This relationship indicates that the effective "de Broglie" wavelength is inversely proportional to the momentum of the particle. For example, an electron at an energy of 10 keV has a wavelength of 0.01 nm, allowing the electron microscope (SEM or TEM) to achieve high resolution images. Other massive particles such as helium, neon, and gallium ions have been used to produce images at resolutions beyond what can be attained with visible light. Such instruments provide nanometer scale imaging, analysis and fabrication capabilities at the expense of system complexity.

References

↑ Born, Max; Emil Wolf (1997). Principles of Optics. Cambridge University Press. ISBN 0-521-63921-2.
↑ Lipson, Lipson and Tannhauser (1998). Optical Physics. United Kingdom: Cambridge. p. 340. ISBN 978-0-521-43047-0.
↑ Fliegel, Karel (December 2004). "Modeling and Measurement of Image Sensor Characteristics" (PDF). Radioengineering 13 (4).
↑ Niek van Hulst (2009). "Many photons get more out of diffraction". Optics & Photonics Focus 4 (1).
↑ Y.Kuznetsova; A.Neumann, S.R.Brueck (2007). "Imaging interferometric microscopy–approaching the linear systems limits of optical resolution". Optics Express 15 (11): 6651–6663. Bibcode:2007OExpr..15.6651K. doi:10.1364/OE.15.006651. PMID 19546975.

External links

Puts, Erwin (September 2003). "Chapter 3: 180 mm and 280 mm lenses" (PDF). Leica R-Lenses. Leica Camera. Archived from the original (PDF) on December 17, 2008. Describes the Leica APO-Telyt-R 280mm f/4, a diffraction-limited photographic lens.

Optical microscopy

Microscope Optical microscopy

Illumination and contrast methods	Bright-field microscopy Köhler illumination Dark field microscopy Phase contrast Quantitative phase-contrast microscopy Differential interference contrast (DIC) Dispersion staining Second harmonic imaging (SHIM) 4Pi microscope Structured illumination Sarfus

Fluorescence methods	Fluorescence microscopy Confocal microscopy Two-photon excitation microscopy Multiphoton microscopy Image deconvolution Total internal reflection fluorescence microscopy (TIRF) Lightsheet microscopy (LSFM/SPIM)

Sub-diffraction limit techniques	Diffraction limit Stimulated emission depletion (STED) Photo-activated localization microscopy (PALM/STORM) Near-field (NSOM/SNOM)

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