Resolution of the optical system. Big encyclopedia of oil and gas. Resolution of optical devices
Resolution(resolving power) optical instruments, characterizes the ability of these devices to provide separate images of two points of an object close to each other. The smallest linear or angular distance between two points, from which their images merge, is called the linear or angular resolution limit. Its reciprocal is usually used as quantity. a measure of resolution. Due to diffraction of light at the edges optical parts even in an ideal optical system (i.e., aberration-free; see Aberrations of optical systems), the image of a point is not a point, but a circle with a central light spot surrounded by rings (alternately dark and light in monochromatic light, iridescently colored in white light) . Diffraction theory makes it possible to calculate the smallest distance resolved by the system, if it is known at what illumination distributions the receiver (eye, photo layer) perceives images separately. According to Rayleigh (1879), images of two points of equal brightness can still be seen separately if the center of the diffraction spot of each of them is intersected by the edge of the first dark ring of the other (Fig.). In the case of self-luminous points emitting incoherent rays, when this Rayleigh criterion is met, the minimum illumination between images of resolved points will be 74% of its maximum value, and the angular distance between the centers of diffraction spots (illuminance maxima) Δφ = 1.21 λ/D where λ is the length waves of light, D is the diameter of the entrance pupil optical system(see Aperture in optics). If f is the focal length of the optical system, then linear quantity Rayleigh resolution limit σ = 1.21 λf/D. The resolution limit of telescopes and spotting scopes is expressed in arcseconds (see Resolving Power of a Telescope); for a wavelength λ ~ 560 nm, corresponding to the maximum sensitivity of the human eye, it is equal to α" = 140/D (D in mm). For photographic lenses, resolution is usually defined as maximum amount separately visible lines per 1 mm image of a standard test object (see Mira) and are calculated using the formula N = 1470ε, where ε is the relative aperture of the lens (see also Resolution of the photographing system; for the resolution of microscopes, see Art. Microscope) . The given relations are valid only for points located on the axis of an ideal optical system. The presence of aberrations and manufacturing errors increases the size of diffraction spots and reduces resolution real systems, which, in addition, decreases with distance from the center of the field of view. The resolution of an optical device R op, which includes an optical system with a resolution of R oc and a light receiver (photolayer, cathode of an electron-optical converter, etc.) with a resolution of R p, is determined by the approximate formula 1/R op = 1/R oc + 1/R p; It follows from it that it is advisable to use only combinations in which R os and R p are quantities of the same order. The resolution of a device can be assessed by its hardware function, which reflects all factors affecting image quality (diffraction, aberration, etc.). Along with assessing image quality by resolution, the method of assessing it using frequency-contrast characteristics is widely used. For the resolution of spectral instruments, see Art. Spectral devices.
Illumination distribution E in the image of two point light sources located so that the angular distance Δφ between the illumination maxima is equal to the angular value Δθ of the radius of the central diffraction spot (Δφ = Δθ - Rayleigh condition).
Using even an ideal optical system (one without defects and aberrations), it is impossible to obtain a stigmatic image point source, which is explained by the wave nature of light. The image of any luminous point in monochromatic light is a diffraction pattern, that is, the point source is displayed as a central bright spot surrounded by alternating dark and light rings.
According to Rayleigh criterion, images of two nearby identical point sources or two nearby spectral lines with equal intensities and identical symmetrical contours are resolvable (separated for perception) if the central maximum of the diffraction pattern from one source (line) coincides with the first minimum of the diffraction pattern from the other (Fig. 265, a ). When the Rayleigh criterion is met, the intensity of the “dip” between the maxima is 80% of the intensity at the maximum, which is sufficient to resolve the lines 1 and 2. If the Rayleigh criterion is violated, then one line is observed (Fig. 265, b).
1. Lens resolution. If light from two distant point sources falls on the lens S 1 and S 2 (for example, stars) with some angular distance , then, due to the diffraction of light waves at the edges of the diaphragm limiting the lens, in its focal plane, instead of two points, maxima are observed, surrounded by alternating dark and light rings (Fig. 266). It can be proven that two nearby stars observed in the lens in monochromatic light are resolvable , if the angular distance between them
where is the wavelength of light, D- lens diameter.
Resolving power (resolving power) of the lens is called the quantity
Where - the smallest angular distance between two points at which they are still resolved by an optical device.
According to the Rayleigh criterion, images of two identical points are resolvable when the central maximum of the diffraction pattern for one point coincides with the first minimum of the diffraction pattern for the other (Fig. 266). It follows from the figure that when the Rayleigh criterion is met, the angular distance between points should be equal , i.e., taking into account (183.1)
Therefore, the lens resolution
i.e., it depends on its diameter and the wavelength of light.
From formula (183.2) it is clear that to increase the resolution of optical instruments, it is necessary either to increase the diameter of the lens or to reduce the wavelength. Therefore, to observe more small parts object using ultraviolet radiation, and the resulting image in in this case observed using a fluorescent screen or recorded on a photographic plate. Even greater resolution could be obtained using x-rays, but it has a high penetrating power and passes through matter without refraction; therefore, in this case it is impossible to create refractive lenses. Electron streams (at certain energies) have approximately the same wavelength as X-rays. Therefore, an electron microscope has a very high resolution.
The resolution of a spectral device is a dimensionless quantity
Where - absolute value the minimum difference in wavelengths of two adjacent spectral lines at which these lines are recorded separately.
2. Resolution of the diffraction grating. Let the maximum T- th order for wavelength 2 is observed at an angle , i.e., according to (180.3), d sin =m 2 . When moving from a maximum to an adjacent minimum, the path difference changes to /N(see (180.4)), where N- number of grid slits. Therefore, the minimum 1 observed at an angle min , satisfies the condition d sin min = m 1 + 1 /N. According to the Rayleigh criterion, = min , i.e. m 2 =m 1 + 1 /N or 2 / ( 2 – 1)=mN. Tax as 1 and 2 are close to each other, i.e. 2 – 1 = then, according to (183.3),
Thus, the resolution of a diffraction grating is proportional to the order m spectrum and number N slots, i.e. when given number gaps increase when moving to large values order m interference. Modern diffraction gratings have a fairly high resolution (up to 210 5).
Dispersion of Light
As already mentioned, light passing through a triangular prism is refracted and, when leaving the prism, deviates from its original direction towards the base of the prism. The amount of beam deflection depends on the refractive index of the prism material, and, as experiments show, the refractive index depends on the frequency of light. The dependence of the refractive index of a substance on the frequency (wavelength) of light is called dispersion. It is very easy to observe the phenomenon of transmission dispersion white light through a prism (Fig. 102). When leaving the prism, white light is decomposed into seven colors: red, orange, yellow, green, blue, indigo, violet. Red light deviates the least, violet light deviates the most. This suggests that glass has the highest refractive index for violet light, and the lowest for red light. Light with different wavelengths propagates in a medium with at different speeds: purple with the smallest, red with the largest, since n= c/v ,
As a result of the passage of light through a transparent prism, an ordered arrangement of monochromatic electromagnetic waves optical range - range.
All spectra are divided into spectra emissions and spectra absorption. The emission spectrum is created by luminous bodies. If a cold, non-emitting gas is placed in the path of the rays incident on the prism, then dark lines appear against the background of the continuous spectrum of the source.
In this case, we obtain the absorption spectrum of the gas. The German physicist G. Kirchhoff (1824-1887) discovered the law according to which the spectral composition of light that is emitted by bodies in a hot state is absorbed by them in a cold state (atoms of a given element absorb those wavelengths that are emitted at high temperatures).
The emission spectra are divided into solid, lined And striped. A continuous spectrum is produced by hot solids and liquids. A line spectrum is a collection of specific spectral lines (on a black background). This spectrum is produced by excited gases in the atomic state. Isolated atoms of a given chemical element emit strictly defined wavelengths. The banded spectrum consists of individual spectral bands separated by dark spaces. Unlike line spectra, striped spectra are created not by atoms, but by molecules that are not bound or weakly bound to each other.
ELECTRONIC THEORY OF LIGHT DISPERSION
From Maxwell’s macroscopic electromagnetic theory it follows that the absolute refractive index of the medium
where is the dielectric constant of the medium, - magnetic permeability. In the optical region of the spectrum for all substances 1, therefore
From formula (186.1) some contradictions with experiment are revealed: the quantity n, being a variable, remains at the same time equal to a certain constant . In addition, the values n, obtained from this expression, do not agree with the experimental values. The difficulties of explaining the dispersion of light from the point of view of Maxwell's electromagnetic theory are eliminated by Lorentz's electronic theory. In Lorentz's theory, light dispersion is considered as a result of the interaction of electromagnetic waves with charged particles that are part of the substance and perform forced oscillations in the alternating electromagnetic field of the wave.
Let us apply the electronic theory of light dispersion for a homogeneous dielectric, formally assuming that light dispersion is a consequence of the dependence from frequency light waves. The dielectric constant of a substance, by definition (see (88.6) and (88.2)), is equal to
Where { - dielectric susceptibility of the medium, 0 - electrical constant, R - instantaneous polarization value. Hence,
(186.2)
those. depends on R. In this case, electronic polarization is of primary importance, i.e. forced oscillations of electrons under the influence of the electric component of the wave field, since for the orientational polarization of molecules the frequency of oscillations in the light wave is very high ( 10 15 Hz).
To a first approximation, we can assume that forced vibrations are performed only by the external electrons most weakly associated with the nucleus - optical electrons. For simplicity, we consider the vibrations of only one optical electron. The induced dipole moment of an electron performing forced oscillations is equal to p=ex, Where e- electron charge, X - displacement of an electron under the influence of the electric field of a light wave. If the concentration of atoms in a dielectric is equal to n 0, then the instantaneous polarization value
From (186.2) and (186.3) we obtain
(186.4)
Consequently, the task comes down to determining the displacement X electron under the influence external field E. We will consider the field of the light wave to be a function of frequency , i.e., varying according to harmonic law: E = E 0cos t.
The equation of forced oscillations of an electron (see §147) for the simplest case (without taking into account the resistance force that determines the absorption of the energy of the incident wave) will be written in the form
Where T, - weight i- th charge.
From expressions (186.8) and (186.9) it follows that the refractive index n depends on frequency external field, i.e., the obtained dependences actually confirm the phenomenon of light dispersion, albeit under the above assumptions, which must be eliminated in the future. From expressions (186.8) and (186.9) it follows that in the region from = 0 to = 0 n 2 is greater than one and increases with increasing (normal variance); at = 0 n 2 = ±; in the area from = 0 to = n 2 is less than one and increases from – to 1 (normal variance). Moving from n 2 k n, we find that the dependence graph n from has the form shown in Fig. 270. This behavior n close 0 - the result of the assumption that there are no resistance forces during electron oscillations. If we take this circumstance into account, then the graph of the function n( ) near 0 will be given by the dashed line AB. Region AB - area of anomalous dispersion ( n decreases as it increases ), other parts of the dependence n from describe the normal variance ( n increases with increasing ).
Russian physicist D. S. Rozhdestvensky (1876-1940) contributed a classic work on the study of anomalous dispersion in sodium vapor. He developed the interference method for very precise measurement refractive index of vapors and experimentally showed that formula (186.9) correctly characterizes the dependence n from , and also introduced a correction to it that takes into account the quantum properties of light and atoms.
If another screen B with a hole is placed between screen A and the light source illuminating it, then a light spot limited by a shadow will appear on screen A (Fig. 319, a and b). The boundary of the shadow can be found geometrically, assuming that light propagates rectilinearly, that is, light rays are straight lines (see Fig. 319, a). However, closer observation shows that the edge of the shadow is not sharp; this is especially noticeable in cases where the hole size is very small compared to the distance
Screen to hole
Then the spot on screen A appears to consist of alternating light and dark rings, gradually turning into each other and also capturing the area of the geometric shadow (Fig. 320, b). This indicates the non-linearity of the propagation of light from the source and the bending of light rays (waves) at the edges of hole B (Fig. 320, a). The described phenomenon of non-linear propagation of light near an obstacle (bending a light beam around an obstacle) is called diffraction of light, and the resulting picture on the screen is called diffraction. When white light is used, the diffraction pattern becomes rainbow-colored.
Let us recall that diffraction is characteristic not only of light, but also of all waves in general (see § 34).
In addition to holes in screens, diffraction is also caused by opaque objects (obstacles) placed in the path of light propagation; it is only necessary that the size of the object be small compared to the distance to the place where the diffraction pattern is observed. In Fig. 321 shows photographs of typical diffraction patterns produced by a round hole a, a rectangular slit by wire b and a screw
Distinct diffraction patterns are obtained in cases where very small obstacles of the order of the light wavelength are in the path of light propagation. It should, however, be emphasized that, contrary to a fairly common belief, the size of the obstacle is not comparable with the wavelength of light. a necessary condition to observe diffraction.
Diffraction patterns often occur in natural conditions. For example, colored rings surrounding a light source, observed through fog or through fogged window glass, are caused by the diffraction of light by tiny water droplets.
Diffraction reveals the wave properties of light and therefore can be explained on the basis of the Huygens-Fresnel principle as follows. Let light from a source fall on screen A through a round hole in screen B (Fig. 322). According to the Huygens-Fresnel principle, each point of the light wave front section (filling the hole) is a secondary light source.
These sources are coherent, so rays (waves) 1 and 2, 3 and 4, etc. emanating from them will interfere with each other. Depending on the magnitude of the difference in the path of the rays on the screen, maximums and minimums of illumination will appear at the points. Thus, on screen A, bright areas will appear in the region of the geometric shadow, and dark areas will appear outside this area, creating the previously described (ring-shaped) diffraction pattern.
Diffraction of light determines the resolution of optical instruments, i.e., the ability of these instruments to produce separate images of small, closely spaced parts (points) of an object. The lens of any optical device must have an entrance hole. Diffraction of light at the entrance hole of the lens inevitably leads to the fact that the images of individual points of the observed object (self-luminous or illuminated) are no longer points, but light disks bordered by dark and light rings. If the points (details) of an object under consideration are close to each other, then their diffraction images (in the focal plane of the lens) can more or less overlap each other (Fig. 323, a).
Two close points 1 and 2 of an object can still be seen separately if the light disks of their diffraction images mutually overlap by no more than the radius of the disk (Fig. 323, b). If the disks overlap by more than a radius (Fig. 323, c), then separate vision of the points becomes impossible; the device no longer separates, or, as they say, does not resolve, such points.
The smallest distance at which two points of an object can still be seen separately is called the resolvable distance. The resolution of an optical device is usually measured by the reciprocal of the resolved distance.
Calculations show that for a microscope the resolvable distance is expressed by the formula
where X is the wavelength of light, the refractive index of the medium located between the object and the lens, and is the aperture angle, i.e., the angle formed by the outer rays of the light beam entering the lens (Fig. 324). The product is called the numerical aperture.
the linear or angular distance between two points, from which their images merge, is called the linear or angular resolution limit. Its reciprocal value usually serves as a quantitative measure Due to light diffraction at the edges of optical parts, even in an ideal optical system (i.e., aberration-free; see Aberrations of optical systems ) the image of a point is not a point, but a circle with a central light spot surrounded by rings (alternately dark and light in monochromatic light , rainbow-colored - in white light ). Diffraction theory allows one to calculate the smallest distance resolved by a system if it is known under what distributions illumination The receiver (eye, photo layer) perceives images separately. According to Rayleigh (1879), images of two points of equal brightness can still be seen separately if the center of the diffraction spot of each of them is intersected by the edge of the 1st dark ring of the other ( rice. ). In the case of self-luminous points emitting incoherent rays, when this Rayleigh criterion is met, the lowest illumination between the images of resolved points will be 74% of its maximum value, and the angular distance between the centers of diffraction spots (illuminance maxima) Dj = 1.21 l ID, where l - wavelength of light, D- diameter of the entrance pupil of the optical system (see. Diaphragm in optics). If f is the focal length of the optical system, then the linear value of the Rayleigh resolution limit s = 1.21 l flD. The resolution limit of telescopes and spotting scopes expressed in arcseconds (see Telescope resolving power ), for wavelength l @ 560 nm, corresponding to the maximum sensitivity of the human eye, it is equal to a" = 140/D ( D V mm). For photographic lenses Resolution (optical) usually defined as the maximum number of separately visible lines per 1 mm images of a standard test object (see. Mira ) and calculated using the formula = 1470e, where e - relative aperture lens (see also Resolution photographing system; O Resolution (optical) microscopes, see Art. Microscope ). The given relations are valid only for points located on the axis of an ideal optical system. The presence of aberrations and manufacturing errors increases the size of diffraction spots and reduces Resolution (optical) real systems, which, in addition, decreases with distance from the center field of view. Resolution (optical) optical device R oops, V which includes an optical system with Resolution (optical) R oc and light receiver (photolayer, cathode electron-optical converter etc.) with Resolution (optical) R n, is determined by the approximate formula 1 /R op = 1/R oc + 1/R n, it follows from it that it is advisable to use only combinations in which R oc and R P - quantities of the same order. Resolution (optical) device can be assessed by its hardware function , reflecting all factors affecting image quality (diffraction, aberrations, etc.). Along with assessing image quality by Resolution (optical) a widely used method for assessing it is using frequency-contrast characteristics. ABOUT Resolution (optical) spectral devices, see Art. Spectral devices.Lit.: Tudorovsky A.I., Theory of optical instruments, 2nd ed., part 1, M. - L., 1948; Landsberg G.S., Optics, 4th ed., M., 1957 (General course of physics, vol. 3); Volosov D.S., Photographic optics, M., 1971.
Article about the word " Resolution (optical)" in the Great Soviet Encyclopedia was read 16229 times
The phenomenon of diffraction puts a limit on the resolving power of many optical instruments and the human eye.
In daylight, the diameter of the pupil, i.e. the diameter D of the hole through which light is diffraction, is approximately 2 mm; let us take the wavelength of light to be equal to Then the angular radius a of the central light diffraction spot when a parallel beam of light hits the pupil of the eye can be determined by formula (15.3):
Thus, as a result of diffraction, an infinitely distant point source is perceived by the eye as a bright spot
with an angular radius of approximately one arc minute. Two luminous points can be perceived by the eye as separate light sources, provided that the angular distance between them exceeds the angular radius of the central diffraction light spot from one point source (Fig. 66). Therefore, the resolution of the human eye is approximately one minute of arc.
When photographing stars using a telescope, the image of the stars on the photographic plate is not pinpoint. This is a consequence of the diffraction of light at the aperture of the telescope lens (Fig. 67). The radius of the central light diffraction spot on a photographic plate can be determined from condition (15.3):
where is the focal length. But in other way,
Expression (15.4) shows that the images of stars on a photographic plate are closer to point-like, the larger the diameter D of the telescope lens and the smaller its focal length F.
Let's estimate the resolution of the world's largest Soviet telescope with a lens diameter of 6 m:
Therefore, with the help of the world's largest optical telescope, it is possible to discern in the sky glowing objects: stars, features on the surface of planets, separated from each other by at least two hundredths of an arcsecond.
The phenomenon of diffraction also limits the resolution of the microscope. Obviously, if in the image constructed by the microscope lens, two luminous points become indistinguishable as a result of the superposition of their diffraction images, then further magnification of the image using an eyepiece cannot make them distinguishable. Therefore, as in the case of determining the resolving power of the eye and telescope, the minimum angular distance between points that can be resolved as separate light sources is approximately equal to the angular radius a of the central bright diffraction spot. According to expression (15.3), the angle is expressed in terms of the lens diameter D and the light wavelength:
Denoting the distance from the object to the microscope lens by (Fig. 68), we obtain the following expression for the minimum linear distance y between two luminous points and B, at which they can be resolved when observed through a microscope:
It can be seen from this that the resolution of a microscope increases with increasing diameter of the microscope lens, with a decrease in the light wavelength and the distance from the lens to the object.
Since the microscope lens must construct a real image, then
Therefore, to reduce the distance, it is necessary to use lenses as short as possible. Increasing the resolution of a microscope lens at a given focal length by increasing the diameter D of the lens is limited by the natural limit:
where is the radius of curvature of the lens. This means that the plano-convex lens, usually used as the first lens of a microscope objective, must be hemispherical.
Since the focal length of a plano-convex lens is determined by the formula
then for the microscope lens we can write the relation:
Taking this into account, the minimum distance at which two luminous points can be located, distinguishable using a microscope, can be expressed as follows:
Taking the refractive index of the glass from which the objective lens is made, we obtain:
Thus, the minimum distance at which two luminous points can be resolved using a microscope with an optimal lens design is approximately equal to the wavelength of the light.
One of possible ways increasing the resolution of an optical microscope is to use short-wave ultraviolet radiation. Since ultraviolet radiation is not perceived by the human eye, but has a strong effect on the photographic plate, the image is photographed, developed and then examined.
