Diffraction-limited system

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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 Abbe diffraction limit for a microscope[edit]

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 diameter

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 which is large compared to most nanostructures or biological cells which have sizes on the order of 1 μm and internal organelles which are much smaller. 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[edit]

You can easily calculate the pixel pitch on a digital camera by dividing the sensor's dimensions in mm, by the number of pixels in the native resolution image. If the diffraction limited spot size is significantly larger than the pixel pitch, the lens is said to have 'empty magnification'. This means that further increasing focal length, at the expense of f/# (such as by using a teleconverter) will not improve resolution in the image. At best, if the teleconverter is perfect, it will get bigger & blurrier in proportion. Conversely, opening the aperture to gather more light, will improve the resolution, in addition to raising the shutter speed.

The numerical aperture page explains that modern, fast, well corrected (i.e. multi-element, computer-designed) lenses are poorly served by the thin-lens approximation, and that f/# can be reasonably defined as 1/(2NA). Conveniently, this means the diffraction limited spot size, on a sensor in air, is just wavelength × f/#. Visible light ranges from 0.35 to 0.7 μm or so, but green, at 0.5 μm, is a good middle value to use. Thus, a perfectly constructed, diffraction limited lens, shooting at f/8 has a maximum resolution of 4 μm at the sensor.

This resolution-limited f/# depends only on the pixel pitch of the sensor. It does not depend on working distance to the subject or focal length.

Interestingly, for extreme macro-photography, this shows that attempting to increase the depth of field by radically reducing the aperture will "cost you" not only in shutter speed or strobe power, but it will also soften the best attainable focus, in the middle of the field.

As examples, we can compute the pixel pitch of the sensors for a full frame "professional" SLR, a "compact" SLR or mirrorless camera, a "respectable" point & shoot, and a sub-compact "toy" camera. As of March of 2014, these are reasonably well represented by the 36 Mpixel Nikon D800 which has 4.88um pixels, the 16 Mpixel Olympus OM-D/E-PL5 which has 3.75um pixels, the 12 Mpixel Canon S120 which has 1.9 μm pixels, and the 16 Mpixel Powershot A4000 which has 1.25 μm pixels.

If we equate the pixel size with the diffraction limited spot size, we can calculate the smallest aperture which is (theoretically) usable without blurring the image. For the D800 it is f/11, for the OM-D it's about f/8.5, for the S120 it is f/4.3 and for the A4000 it is f/2.8. (On the A4000, this is faster than the built in lens is capable of achieving, so it is always operating with 'empty' magnification)

Of course, computing diffraction limited resolution is always an optimistic exercise, and occasionally wildly optimistic. Like EPA highway mileage, real world performance is often significantly worse.

Obtaining higher resolution[edit]

There are techniques for producing images that appear to have higher resolution than allowed by simple use of diffraction-limited optics.[3] 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[edit]

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.[4] 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[edit]

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[edit]

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.

Other waves[edit]

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.

See also[edit]

References[edit]

  1. ^ Born, Max; Emil Wolf (1997). Principles of Optics. Cambridge University Press. ISBN 0-521-63921-2. 
  2. ^ Lipson, Lipson and Tannhauser (1998). Optical Physics. United Kingdom: Cambridge. p. 340. ISBN 978-0-521-43047-0. 
  3. ^ Niek van Hulst (2009). "Many photons get more out of diffraction". Optics & Photonics Focus 4 (1). 
  4. ^ 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. 

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