Focal length (f) and field of view (FOV) of a lens are inversely proportional. For a standard rectilinear lens, F O V = 2 arctan ⁡ ( x 2 f ) {\textstyle \mathrm {FOV} =2\arctan {\left({x \over 2f}\right)}} , where x is the width of the film or imaging sensor.

In order to increase the resolution, d = λ/(2NA), the specimen must be viewed using either a shorter wavelength (λ) of light or through an imaging medium with a relatively high refractive index or with optical components which have a high NA (or, indeed, a combination of all of these factors).

Principalfocus ofconcavelensis real or virtual

Taking all of the above theories into consideration, it is clear that there are a number of factors to consider when calculating the theoretical limits of resolution. Resolution is also dependent on the nature of the sample. Let’s look at calculating resolution using the Abbe diffraction limit, Rayleigh Criterion, and also FWHM.

Due to the popularity of the 35 mm standard, camera–lens combinations are often described in terms of their 35 mm-equivalent focal length, that is, the focal length of a lens that would have the same angle of view, or field of view, if used on a full-frame 35 mm camera. Use of a 35 mm-equivalent focal length is particularly common with digital cameras, which often use sensors smaller than 35 mm film, and so require correspondingly shorter focal lengths to achieve a given angle of view, by a factor known as the crop factor.

As s1 is decreased, s2 must be increased. For example, consider a normal lens for a 35 mm camera with a focal length of f = 50 mm. To focus a distant object (s1 ≈ ∞), the rear principal plane of the lens must be located a distance s2 = 50 mm from the film plane, so that it is at the location of the image plane. To focus an object 1 m away (s1 = 1,000 mm), the lens must be moved 2.6 mm farther away from the film plane, to s2 = 52.6 mm.

However, even taking all of these factors into consideration, the possibilities with a real microscope are still somewhat limited due to the complexity of the whole system, transmission characteristics of glass at wavelengths below 400 nm, and the challenge to achieve a high NA in the complete microscope system. Lateral resolution in an ideal optical microscope is limited to around 200 nm, whereas axial resolution is around 500 nm (examples of resolution limits are given below).

The Rayleigh Criterion is a slightly refined formula based on Abbe’s diffraction limits:  R = 1.22λ/(NAobj + NAcond) where λ is the wavelength of light used to image a specimen. NAobj is the NA of the objective. NAcond is the NA of the condenser. The value ‘1.22’ is a constant. This is derived from Rayleigh’s work on Bessel Functions. These are used for calculating problems in systems such as wave propagation.

The optical power of a lens or curved mirror is a physical quantity equal to the reciprocal of the focal length, expressed in metres. A dioptre is its unit of measurement with dimension of reciprocal length, equivalent to one reciprocal metre, 1 dioptre = 1 m−1. For example, a 2-dioptre lens brings parallel rays of light to focus at 1⁄2 metre. A flat window has an optical power of zero dioptres, as it does not cause light to converge or diverge.[10]

Camera lens focal lengths are usually specified in millimetres (mm), but some older lenses are marked in centimetres (cm) or inches.

Infinitefocus lens

An Airy disc is the optimally focused point of light which can be determined by a circular aperture in a perfectly aligned system limited by diffraction. Viewed from above (Figure 1), this appears as a bright point of light around which are concentric rings or ripples (more correctly known as an Airy Pattern).

Firstly, it should be remembered that: NA = n(sinα) where n is the refractive index of the imaging medium and α is half of the angular aperture of the objective. The maximum angular aperture of an objective is around 144º. The sine of half of this angle is 0.95. If using an immersion objective with oil which has a refractive index of 1.52, the maximum NA of the objective will be 1.45. If using a ‘dry’ (non-immersion) objective the maximum NA of the objective will be 0.95 (as air has a refractive index of 1.0).

1 f = ( n − 1 ) ( 1 R 1 − 1 R 2 + ( n − 1 ) d n R 1 R 2 ) , {\displaystyle {\frac {1}{f}}=(n-1)\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right),} where n is the refractive index of the lens medium. The quantity ⁠1/f⁠ is also known as the optical power of the lens.

For a thick lens (one which has a non-negligible thickness), or an imaging system consisting of several lenses or mirrors (e.g. a photographic lens or a telescope), there are several related concepts that are referred to as focal lengths:

Principalfocus of lensdiagram

Also in the year 1835, he published a paper in the Transactions of the Cambridge Philosophical Society entitled ‘On the Diffraction of an Object-Glass with Circular Aperture’ [1]. Airy wrote this paper very much from the view of an astronomer and in it he describes “the form and brightness of the rings or rays surrounding the image of a star as seen in a good telescope”. Despite writing in a different scientific field, these observations are relevant to other optical systems including microscopes.

In the sign convention used here, the value of R1 will be positive if the first lens surface is convex, and negative if it is concave. The value of R2 is negative if the second surface is convex, and positive if concave. Sign conventions vary between different authors, which results in different forms of these equations depending on the convention used.

The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning; it is simply the inverse of the system's optical power.

Principalfocus of lens

In microscopy, the term “resolution” is used to describe the ability of a microscope to distinguish details of a specimen or sample. In other words, the minimum distance between 2 distinct points of a specimen where they can still be seen by the observer or microscope camera as separate entities. Resolution is intrinsically linked to the numerical aperture (NA) of a microscope’s optical components, like the objective lens, as well as the wavelength of light used. This article covers some of the history behind resolution concepts and explains each one using relatively simple terminology.

Partsofa convexlens

Rayleigh built upon and expanded the work of George Airy and invented the theory of the ‘Rayleigh criterion’ in 1896 [3]. The Rayleigh criterion defines the limit of resolution in a diffraction-limited system, in other words, when two points of light are distinguishable or resolved from each other.

The numerical aperture (NA) is related to the refractive index (n) of a medium through which light passes as well as the angular aperture (α) of a given objective (NA = n sinα). The resolution of an optical microscope is not solely dependent on the NA of an objective, but the NA of the whole system, taking into account the NA of the microscope condenser. More image detail will be resolved in a microscope system in which all of the optical components are correctly aligned, have a relatively high NA value and are working harmoniously with each other. Resolution is also related to the wavelength of light which is used to image a specimen; light of shorter wavelengths are capable of resolving greater detail than longer wavelengths.

PartsofalensPhysics

Abbe’s diffraction formula for lateral (XY) resolution is:  d = λ/(2NA) where λ is the wavelength of light used to image a specimen. If using a green light of 514 nm and an oil-immersion objective with an NA of 1.45, then the (theoretical) limit of resolution will be 177 nm.

When a photographic lens is set to "infinity", its rear principal plane is separated from the sensor or film, which is then situated at the focal plane, by the lens's focal length. Objects far away from the camera then produce sharp images on the sensor or film, which is also at the image plane.

For a thin lens in air, the focal length is the distance from the center of the lens to the principal foci (or focal points) of the lens. For a converging lens (for example a convex lens), the focal length is positive and is the distance at which a beam of collimated light will be focused to a single spot. For a diverging lens (for example a concave lens), the focal length is negative and is the distance to the point from which a collimated beam appears to be diverging after passing through the lens.

There are 3 mathematical concepts which need to be taken into consideration when dealing with resolution: Abbe’s diffraction limit, Airy discs, and the Rayleigh criterion. Each of these are covered below in chronological order.

Principalfocus ofconvex mirror

Abbe’s diffraction formula for axial (Z) resolution is:  d = 2λ/(NA)2 and again, if we assume a wavelength of 514 nm to observe a specimen with an objective having an NA value of 1.45, then the axial resolution will be 488 nm.

Using the theory of Airy discs, if the diffraction patterns from two single Airy discs do not overlap, then they are easily distinguishable, ‘well resolved’ and are said to meet the Rayleigh criterion. When the center of one Airy disc is directly overlapped by the first minimum of the diffraction pattern of another, they can be considered to be ‘just resolved’ and still distinguishable as two separate points of light (Figure 2, mid). If the Airy discs are closer than this, then they do not meet the Rayleigh criterion and are ‘not resolved’ as two distinct points of light.

To achieve the maximum theoretical resolution of a microscope system, each of the optical components should be of the highest NA available (taking into consideration the angular aperture). In addition, using a shorter wavelength of light to view the specimen will increase the resolution. Finally, the whole microscope system should be correctly aligned.

The focal length of a thin convex lens can be easily measured by using it to form an image of a distant light source on a screen. The lens is moved until a sharp image is formed on the screen. In this case ⁠1/u⁠ is negligible, and the focal length is then given by

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A lens with a focal length about equal to the diagonal size of the film or sensor format is known as a normal lens; its angle of view is similar to the angle subtended by a large-enough print viewed at a typical viewing distance of the print diagonal, which therefore yields a normal perspective when viewing the print;[9] this angle of view is about 53 degrees diagonally. For full-frame 35 mm-format cameras, the diagonal is 43 mm and a typical "normal" lens has a 50 mm focal length. A lens with a focal length shorter than normal is often referred to as a wide-angle lens (typically 35 mm and less, for 35 mm-format cameras), while a lens significantly longer than normal may be referred to as a telephoto lens (typically 85 mm and more, for 35 mm-format cameras). Technically, long focal length lenses are only "telephoto" if the focal length is longer than the physical length of the lens, but the term is often used to describe any long focal length lens.

Determining the focal length of a concave lens is somewhat more difficult. The focal length of such a lens is defined as the point at which the spreading beams of light meet when they are extended backwards. No image is formed during such a test, and the focal length must be determined by passing light (for example, the light of a laser beam) through the lens, examining how much that light becomes dispersed/ bent, and following the beam of light backwards to the lens's focal point.

As already mentioned, the FWHM can be measured directly from the PSF or calculated using: RFWHM = 0.51λ/(NA). Again using a light wavelength of 514 nm and an objective with an NA of 1.45, then theoretical resolution will be 181 nm. This value is very close to the lateral resolution calculated just above from the Abbe diffraction limit.

These theoretical resolution values, derived from physical and mathematical assumptions, are estimates. They assume perfect imaging systems and a point light source in a vacuum or a completely homogeneous material as the sample or specimen. Of course, this assumption is almost never the case in real life, as many samples or specimens are heterogeneous. Because there is only a finite amount of light transmitting through the sample or reflecting from its surface, the measurable resolution depends significantly on the signal-to-noise ratio (SNR).

For a spherically-curved mirror in air, the magnitude of the focal length is equal to the radius of curvature of the mirror divided by two. The focal length is positive for a concave mirror, and negative for a convex mirror. In the sign convention used in optical design, a concave mirror has negative radius of curvature, so

The corresponding front focal distance is:[6] FFD = f ( 1 + ( n − 1 ) d n R 2 ) , {\displaystyle {\mbox{FFD}}=f\left(1+{\frac {(n-1)d}{nR_{2}}}\right),} and the back focal distance: BFD = f ( 1 − ( n − 1 ) d n R 1 ) . {\displaystyle {\mbox{BFD}}=f\left(1-{\frac {(n-1)d}{nR_{1}}}\right).}

In most photography and all telescopy, where the subject is essentially infinitely far away, longer focal length (lower optical power) leads to higher magnification and a narrower angle of view; conversely, shorter focal length or higher optical power is associated with lower magnification and a wider angle of view. On the other hand, in applications such as microscopy in which magnification is achieved by bringing the object close to the lens, a shorter focal length (higher optical power) leads to higher magnification because the subject can be brought closer to the center of projection.

John William Strutt, 3rd Baron Rayleigh (1842-1919) was an English physicist and a prolific author. During his lifetime, he wrote an astonishing 466 publications including 430 scientific papers. He wrote on a huge range of topics as diverse as bird flight, psychical research, acoustics and in 1895, he discovered argon (Ar) for which he was later awarded the Nobel prize for physics in 1904.

A more practical approach for resolution is the full width at half maximum (FWHM) intensity of an optically unresolved structure [4,5]. This value is relatively easy to measure with a microscope and has become a generally accepted parameter for comparison purposes. The theoretical value for the FWHM is RFWHM = 0.51λ/(NA) which is approximately λ/(2NA). So the FWHM as a resolution parameter is very close to Abbe’s diffraction limit, but also can be measured from microscope image data. For calibration or resolution-limit measurements, often beads or colloids of various diameters are imaged and measured.

Secondaryfocus of lens

Taking the NA of the condenser into consideration, air (with a refractive index of 1.0) is generally the imaging medium between the condenser and the slide. Assuming the condenser has an angular aperture of 144º then the NAcond value will equal 0.95.

For the case of a lens of thickness d in air (n1 = n2 = 1), and surfaces with radii of curvature R1 and R2, the effective focal length f is given by the Lensmaker's equation:[5]

For an optical system in a medium other than air or vacuum, the front and rear focal lengths are equal to the EFL times the refractive index of the medium in front of or behind the lens (n1 and n2 in the diagram above). The term "focal length" by itself is ambiguous in this case. The historical usage was to define the "focal length" as the EFL times the index of refraction of the medium.[2][4] For a system with different media on both sides, such as the human eye, the front and rear focal lengths are not equal to one another, and convention may dictate which one is called "the focal length" of the system. Some modern authors avoid this ambiguity by instead defining "focal length" to be a synonym for EFL.[1]

George Biddell Airy (1801-1892) was an English mathematician and astronomer. By the 1826 (aged 25) he was appointed professor of mathematics at Trinity College and two years later, he was appointed professor of astronomy at the new Cambridge Observatory. From 1835 to 1881 he was the ‘Astronomer Royal’ and even has a lunar and Martian crater named in his honor.

Ernst Karl Abbe (1840-1905) was a German mathematician and physicist. In 1866 he met Carl Zeiss and together they founded what was known as the ‘Zeiss Optical Works’, now known as Zeiss. In addition, he also co-founded Schott Glassworks in 1884. Abbe was also the first person to define the term numerical aperture. In 1873, Abbe published his theory and formula which explained the diffraction limits of the microscope [2]. Abbe recognized that specimen images are composed of a multitude of overlapping, multi-intensity, diffraction-limited points (or Airy discs).

The distinction between front/rear focal length and EFL is important for studying the human eye. The eye can be represented by an equivalent thin lens at an air/fluid boundary with front and rear focal lengths equal to those of the eye, or it can be represented by a different equivalent thin lens that is totally in air, with focal length equal to the eye's EFL.

The main benefit of using optical power rather than focal length is that the thin lens formula has the object distance, image distance, and focal length all as reciprocals. Additionally, when relatively thin lenses are placed close together their powers approximately add. Thus, a thin 2.0-dioptre lens placed close to a thin 0.5-dioptre lens yields almost the same focal length as a single 2.5-dioptre lens.

When a lens is used to form an image of some object, the distance from the object to the lens u, the distance from the lens to the image v, and the focal length f are related by

For an optical system in air the effective focal length, front focal length, and rear focal length are all the same and may be called simply "focal length".

The three-dimensional (3D) representation of the Airy pattern, as illustrated in the right half of Figure 1, is also known as the ‘point-spread function’ (PSF) of an optical instrument which has no appreciable aberration.

As stated above, the shorter the wavelength of light used to image a specimen, then the more the fine details are resolved. So, if using the shortest wavelength of visible light, 400 nm, with an oil-immersion objective having an NA of 1.45 and a condenser with an NA of 0.95, then R would equal 203 nm.

To render closer objects in sharp focus, the lens must be adjusted to increase the distance between the rear principal plane and the film, to put the film at the image plane. The focal length f, the distance from the front principal plane to the object to photograph s1, and the distance from the rear principal plane to the image plane s2 are then related by:

The diffraction pattern is determined by the wavelength of light and the size of the aperture through which the light passes. The central point of the Airy disc contains approximately 84% of the luminous intensity with the remaining 16% in the diffraction pattern around this point. There are of course many points of light in a specimen as viewed with a microscope, and it is more appropriate to think in terms of numerous Airy patterns as opposed to a single point of light as described by the term ‘Airy disc’.

The focal length of a lens determines the magnification at which it images distant objects. It is equal to the distance between the image plane and a pinhole that images distant objects the same size as the lens in question. For rectilinear lenses (that is, with no image distortion), the imaging of distant objects is well modelled as a pinhole camera model.[7] This model leads to the simple geometric model that photographers use for computing the angle of view of a camera; in this case, the angle of view depends only on the ratio of focal length to film size. In general, the angle of view depends also on the distortion.[8]

If using a green light of 514 nm, an oil-immersion objective with an NA of 1.45, condenser with an NA of 0.95, then the (theoretical) limit of resolution will be 261 nm.