(c) How big a spot would be illuminated on the Moon, neglecting atmospheric effects? (This might be done to hit a corner reflector to measure the round-trip time and, hence, distance.) Explicitly show how you follow the steps in Chapter Problem-Solving Strategies for Wave Optics.

Aspheric lenses minimize spherical aberrations and offer a smaller footprint resulting in lighter, more compact optical systems.

The distance s  between two objects a distance r away and separated by an angle θ is s = r θ, by definition of an angle in radians.

There are obvious differences in imaging principles between short-wave infrared and medium-long wave infrared detectors.Short wave infraredusesreflected ...

14: Covid-19 virus has a size of 88 nm.  An optical compound microscope such as that found in most first year college labs has an objective lens with a diameter of 2.0 cm and the virus is on a slide that is being viewed from a distance of 0.50 cm.  If it is being viewed in light with a wavelength of 555 nm.  a) what is the resolution in a) metres b) nanometres of this microscope?  Can you see the virus using this microscope? Yes or no?

Consider diffraction limits for an electromagnetic wave interacting with a circular object. Construct a problem in which you calculate the limit of angular resolution with a device, using this circular object (such as a lens, mirror, or antenna) to make observations. Also calculate the limit to spatial resolution (such as the size of features observable on the Moon) for observations at a specific distance from the device. Among the things to be considered are the wavelength of electromagnetic radiation used, the size of the circular object, and the distance to the system or phenomenon being observed.

The effect of spherical aberration manifests itself in two ways: the center of the image remains more in focus than the edges, and the intensity of the edges falls relative to that of the center. This defect appears in both on-axis and off-axis image points.

Apr 3, 2020 — Short focal lengths make objects look farther apart whilst longer lengths compress distances and make objects seem much closer together.

10: distance = 7.9 x 10-5 m     Now to a ratio 1 inch=2.54 cm.  2.54 x10-2 m /  7.9 x 10-5 m  = 321 dots per inch.  Common. https://computer.howstuffworks.com/printer-ink3.htm states that common resolutions are 300-600 dpi.

6: angle =2.24 x10-4 radians = x/distance so distance = 5.81 x 103 m = 5.81 km  c) s= 1.79 x10-4 m = 0.179 mm. Take a ruler and hold it away from you.  Most of you can easily see the 1 mm separation so 0.2 mm makes sense.

Spherical aberration definitionlens

Light diffracts as it moves through space, bending around obstacles, interfering constructively and destructively. While this can be used as a spectroscopic tool—a diffraction grating disperses light according to wavelength, for example, and is used to produce spectra—diffraction also limits the detail we can obtain in images. The figure below shows the effect of passing light through a small circular aperture. Instead of a bright spot with sharp edges, a spot with a fuzzy edge surrounded by circles of light is obtained. This pattern is caused by diffraction similar to that produced by a single slit. Light from different parts of the circular aperture interferes constructively and destructively. The effect is most noticeable when the aperture is small, but the effect is there for large apertures, too.

The NA for a lens is NA = n sinα, where n is the index of refraction of the medium between the objective lens and the object at point P.

where d is the distance between the specimen and the objective lens, and we have used the small angle approximation (i.e., we have assumed that x is much smaller than d), so that tan θ ≈  sinθ ≈  θ when is in radians.

13: Microscopes:  x=1.22 λ (d/D) = 2.01×10-6 m = 2.03 microns which is much less than the 20 micron hair so it will be able to resolve it.

Specimens mounted in Canada balsam or similar mounting media that have a refractive index approximating that of the cover glass are not prone to spherical aberration errors. However, this is not true for specimens mounted in physiological saline or other aqueous media with refractive indices significantly different from the cover glass. Even when focusing through thin layers of water only a few microns thick, significant aberrations are encountered that can induce dramatic asymmetries into the point spread function causing a non-uniform distribution above and below the focal plane. This concept is explored in the interactive tutorial linked below.

where n and n' represent the refractive index of air and the glass comprising the lens, respectively, s and s' are the object and image distance, and r is the radius of curvature of the lens. This expression determines the relative locations of images formed by the curved surface of a lens having radius r sandwiched between media of refractive indices n and n'. A refinement of this equation is often referred to as a higher-order (first, second, or third) correction by including terms in the cube of the aperture angle, resulting in a more refined calculation. Departure from an ideal spherical wave is expressed in terms of fractions of a wave, where a single wave is equal to the average wavelength of the illuminating light. This deviation is termed the optical path difference, which must be less than one-quarter wavelength before a diffraction limited objective can be considered aberration-free.

9: (a) Yes. Should easily be able to discern.  (b) The fact that it is just barely possible to discern that these are separate bodies indicates the severity of atmospheric aberrations.   We do not have perfect skies.

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Diffraction is not only a problem for optical instruments but also for the electromagnetic radiation itself. Any beam of light having a finite diameter D  and a wavelength λ exhibits diffraction spreading. The beam spreads out with an angle θ  given by the equation θ = 1.22 λ/D. Take, for example, a laser beam made of rays as parallel as possible (angles between rays as close to θ = 0o as possible) instead spreads out at an angle θ = 1.22 λ/D, where D is the diameter of the beam and λ is its wavelength. This spreading is impossible to observe for a flashlight, because its beam is not very parallel to start with. However, for long-distance transmission of laser beams or microwave signals, diffraction spreading can be significant (see Figure 5). To avoid this, we can increase D. This is done for laser light sent to the Moon to measure its distance from the Earth. The laser beam is expanded through a telescope to make D much larger and θ smaller.

(b) In actuality, it is just barely possible to discern that Pluto and Charon are separate bodies using an Earth-based telescope. What are the reasons for this?

Douglas College Physics 1207 Copyright © August 22, 2016 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

Spherical aberrationexample

b) What is the minimum diameter mirror on a telescope that would allow you to see details as small as 5.00 metres?  This is what you would need to see evidence of the Lunar Landing.

It is also possible for a user to inadvertently introduce spherical aberration into a well-corrected system. For example, when using high magnification, high numerical aperture dry objectives, the correct thickness of the cover glass (suggested to be 0.17 millimeters) is critical. Figure 3 illustrates the changes in half-width of the intensity distribution curve with changes in cover glass thickness. Even with high quality cover glasses having a tolerance of ±10 micrometers, the half-width changes by more than a factor of two. As the objective numerical aperture is increased (above a value of 0.5), particularly with dry and water immersion lenses, selection of cover glasses for the correct thickness is particularly important.

Most of the discrepancy in focal points arises from approximations of the equivalency of sine and tangent values of respective angles made to the Gaussian lens equation for a spherical refracting surface:

What isspherical aberrationin Physics

13: A basic simple magnification microscope is being used in light with a wavelength of 555 nm.  It has an objective lens with a diameter of 5.00 cm. This is the aperture.  The specimen is 15.0 cm away from the objective lens.  What is the resolution in meters? In micrometers which are also called microns?  The width of a human hair is 20 microns.  Will this microscope be able to resolve human hair?

The primary mirror of the orbiting Hubble Space Telescope has a diameter of 2.40 m. Being in orbit, this telescope avoids the degrading effects of atmospheric distortion on its resolution. (a) What is the angle between two just-resolvable point light sources (perhaps two stars)? Assume an average light wavelength of 550 nm. (b) If these two stars are at the 2 million light year distance of the Andromeda galaxy, how close together can they be and still be resolved? (A light year, or ly, is the distance light travels in 1 year.)

Spherical aberrations are very important in terms of the resolution of the lens because they affect the coincident imaging of points along the optical axis and degrade the performance of the lens, which will seriously affect specimen sharpness and clarity. These lens defects can be reduced by limiting the outer edges of the lens from exposure to light using diaphragms and also by utilizing aspherical lens surfaces within the system. However, a consequence of reducing aperture size in the microscope optical system is a concurrent reduction in the amount of light entering the system. Spherical aberration is usually corrected by employing glass elements (lens doublets or triplets) cemented together. The glass elements are designed with different shapes of convexity and/or concavity to insure that the peripheral rays and axial rays, especially at the outer area of the field of view, are brought into common focus.

What is seen in the microscope is an image made by focusing the peripheral rays surrounded by the unfocused image of rays traveling through the central portion of the lens (or visa versa). This is one of the most serious resolution artifacts because the image of the specimen is spread out rather than being in sharp focus. The best focus, in an imperfectly or non-corrected lens, will be somewhere between the focal planes of the peripheral and axial rays, an area known as the disc of least confusion (illustrated as a point on the optical axis in the tutorial figure). Light rays refracted by the rim of the lens or pupil (peripheral rays) have the shortest focal length and produce the smallest image, whereas those that intersect at the paraxial focal point (axial rays) have begun to spread and do not represent the "best" focus.

One of the consequences of diffraction is that the focal point of a beam has a finite width and intensity distribution. Consider focusing when only considering geometric optics, shown in Figure 7(a). The focal point is infinitely small with a huge intensity and the capacity to incinerate most samples irrespective of the NA of the objective lens. For wave optics, due to diffraction, the focal point spreads to become a focal spot (see Figure 7(b)) with the size of the spot decreasing with increasing NA. Consequently, the intensity in the focal spot increases with increasing NA. The higher the NA, the greater the chances of photodegrading the specimen. However, the spot never becomes a true point.

As I write this in March 2020, Covid-19 is a major concern as we do not yet have a vaccine.  Coronavirus are about 82-94 nm in diameter, excluding the spikes. [1]

5: A telescope can be used to enlarge the diameter of a laser beam and limit diffraction spreading. The laser beam is sent through the telescope in opposite the normal direction and can then be projected onto a satellite or the Moon.

In most biology laboratories, resolution is presented when the use of the microscope is introduced. The ability of a lens to produce sharp images of two closely spaced point objects is called resolution. The smaller the distance x by which two objects can be separated and still be seen as distinct, the greater the resolution. The resolving power of a lens is defined as that distance x. An expression for resolving power is obtained from the Rayleigh criterion. In Figure 6(a) we have two point objects separated by a distance x. According to the Rayleigh criterion, resolution is possible when the minimum angular separation is

The tutorial illustrates an exaggerated view of three hypothetical monochromatic light rays passing through a convex lens and converging on a series of focal points that lie in a progression along the optical axis (see the Ray Trace Diagram). Changes to the shape of the lens with corresponding adjustments to the focal point position(s) can be made by utilizing the Lens Shape slider. Refraction of peripheral rays at the edge of the lens is greatest followed by those in the middle and then the rays in the center. The larger refraction by the outermost rays results in a focal point (focal point 1; see Figure 1) that occurs in front of the disc of least confusion and the focal points produced by rays passing closer to the center of the lens (focal points 2 in the center and 3, at the paraxial focal plane; Figure 1). Also illustrated in Figure 1 is a measure of the transverse spherical aberration, defined as the distance from the optical axis at which the peripheral rays intersect the plane of paraxial focus. As is evident in the figure, transverse aberration is measured in the plane of the image and is useful as an indicator of image blur.

If you want to learn more about this and other topics, visit The Microscope Organization’s website at https://microscope-microscope.org/microscope-info/numerical-aperture/

As the objective numerical aperture is increased, changes in cover glass thickness or refractive index become critical, particularly with high magnification dry objectives where small changes in tube length quickly lead to inferior images. Although spherical aberration can be corrected to almost undetectable limits for visual observation with all types of objectives, the optical specification for any given lens must be fulfilled. For oil-immersion objectives having high numerical apertures, this usually means using a cover glass having a 0.17 millimeter thickness and immersion oil with a refractive index of 1.5180 (± 0.0004) at wavelengths of 546 and 589 nanometers. Complicating these conditions is the fact that for almost all materials, refractive index is a function of both wavelength and temperature. In cases where the exact properties of the cover glass and oil are specified, microscope manufacturers can correct spherical aberration for several values of wavelength.

The focal length of a mirror and a lens can be calculated using 1/do + 1/di = 1/f, where do is the object distance, di is the image distance, and f is the focal ...

(b) Take your result to be the practical limit for the eye. What is the greatest possible distance a car can be from you if you can resolve its two headlights, given they are 1.30 m apart?

15: What wavelength of electromagnetic radiation do you need to view the a Covid-19 virus particle with a size of 82 nanometres? Assume the diameter of the objective lens is 2.0 mm and it is being viewed from 3.0 cm away.

Until recent years, achromats were corrected spherically only for green light, although they were corrected chromatically for two wavelengths. Also, apochromats were corrected spherically for two wavelengths, blue and green, but were corrected chromatically for three wavelengths. The highest-quality modern microscope objectives address spherical aberrations in a number of ways including special lens-grinding techniques, improved glass formulations, and better control of optical pathways through use of multiple-lens elements. Currently, the highest quality objectives, planapochromats, are spherically corrected for four wavelengths, as are planfluorites (but not to quite as close a tolerance).

(a) If this is done with the Mount Wilson telescope, producing a 2.54-m-diameter beam of 633-nm light, what is the minimum angular spread of the beam?

(b) Neglecting atmospheric effects, what is the size of the spot this beam would make on the Moon, assuming a lunar distance 3.84 x 108  m?

7: a) What is the minimum diameter mirror on a telescope that would allow you to see details as small as 5.00 km on the Moon some 384,000 km away? Assume an average wavelength of 550 nm for the light received.   Find the angle in radians you need first.

One of the mechanisms used to eliminate spherical aberration in oil immersion objectives is to design the optics around specific pairs of conjugate points using a hemispherical and meniscus lens at the front of the objective. As illustrated in Figure 2, for a specimen observed at position P and surrounded by immersion oil of refractive index n, there exists a conjugate point (P(1)) to eliminate spherical aberration in the first lens element (the hemispherical lens). In this case, light rays emanating from point P leave the surface of the hemispherical front lens as if they originated at point P(1). The meniscus lens is ground with a surface radius centered on point P to form a second conjugate pair (P(1) and P(2)). Thus, light from the specimen a point P ultimately exits the meniscus lens as if it originated at point P(2), eliminating spherical aberration for the lens combination.

2:  What is the smallest detail that could be observed on the Moon when it is 384,000 km away from the Hubble Telescope?  Assume the angular resolution found for the Hubble Telescope is the same as explained in Example 1 — the Hubble Telescope has a mirror diameter of 2.40 metres and the wavelength of the light is 550.0 nm.

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3: Diffraction spreading for a flashlight is insignificant compared with other limitations in its optics, such as spherical aberrations in its mirror. To show this, calculate the minimum angular spreading of a flashlight beam that is originally 5.00 cm in diameter with an average wavelength of 600 nm.

4: (a) What is the minimum angular spread of a 633-nm wavelength He-Ne laser beam that is originally 1.00 mm in diameter?

The most serious of the classical Seidel monochromatic lens aberrations that occurs with microscope objectives, spherical aberration, causes the specimen image to appear hazy or blurred and slightly out of focus. Ideally, an aberration-free objective converts a plane wavefront into a spherical wavefront, directing all light waves refracted by the lens to a common focal point in the center of the sphere to produce a perfect image.

Diamond turning, also known as diamond machining or diamond cutting, is a precision machining process used to produce high-quality optical surfaces and ...

Spherical aberration definitionphotography

Draw two lines on a white sheet of paper (several mm apart). How far away can you be and still distinguish the two lines? What does this tell you about the size of the eye’s pupil? Can you be quantitative? (The size of an adult’s pupil is discussed in another chapter.)

The answer in part (b) indicates that two stars separated by about half a light year can be resolved. The average distance between stars in a galaxy is on the order of 5 light years in the outer parts and about 1 light year near the galactic centre. Therefore, the Hubble can resolve most of the individual stars in Andromeda galaxy, even though it lies at such a huge distance that its light takes 2 million years for its light to reach us. Figure 4 shows another mirror used to observe radio waves from outer space.

Another way to look at this is by re-examining the concept of Numerical Aperture (NA) discussed in the previous chapter that deal with the lenses in a microscope, but did not talk about the wave light nature of light.   Chapter Microscopes. There, NA is a measure of the maximum acceptance angle at which the fibre will take light and still contain it within the fibre. Figure 6(b) shows a lens and an object at point P. The NA here is a measure of the ability of the lens to gather light and resolve fine detail. The angle subtended by the lens at its focus is defined to be  θ = 2 α. From the figure and again using the small angle approximation, we can write

1: A beam of light always spreads out. Why can a beam not be created with parallel rays to prevent spreading? Why can lenses, mirrors, or apertures not be used to correct the spreading?

b)  angle = 1.30 x 10-8 radians  D = 51.3 m.   I often get asked by the general public why they cannot see the Lunar Landing Site with their telescope which is usually about 2 inches in diameter. This is why.

There are many situations in which diffraction limits the resolution. The acuity of our vision is limited because light passes through the pupil, the circular aperture of our eye. Be aware that the diffraction-like spreading of light is due to the limited diameter of a light beam, not the interaction with an aperture. Thus light passing through a lens with a diameter D shows this effect and spreads, blurring the image, just as light passing through an aperture of diameter D does. So diffraction limits the resolution of any system having a lens or mirror. Telescopes are also limited by diffraction, because of the finite diameter D  of their primary mirror.

10: When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh’s criterion. Take the pupil of the eye to be 3.0 mm and the distance from the paper to the eye of 35 cm; find the minimum separation of two dots such that they cannot be resolved. How many dots per inch (dpi) does this correspond to?

Spherical aberrationand chromaticaberration

The Rayleigh criterion stated in the equation θ = 1.22 λ/D gives the smallest possible angle θ between point sources, or the best obtainable resolution. Once this angle is found, the distance between stars can be calculated, since we are given how far away they are.

9: (a) The planet Pluto and its Moon Charon are separated by 19,600 km. Neglecting atmospheric effects, should the 5.08-m-diameter Mount Palomar telescope be able to resolve these bodies when they are 4.50 x 109  km from Earth? Assume an average wavelength of 550 nm.

Spherical aberration definitioneyewiki

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What isspherical aberrationin lenses

where  is the wavelength of light (or other electromagnetic radiation) and D is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression,  θ has units of radians.

The tutorial initializes with an image of the specimen (as seen through the microscope) appearing in a window on the left-hand side of the applet. Beneath the image window is a pull-down menu labeled Choose A Specimen, used to select a new specimen. The Lens Shape slider is designed to control the tutorial by introducing an increasing amount of spherical aberration into the optical system. Moving the slider to the right also induces changes corresponding to the introduction of spherical aberration into the Airy diffraction pattern shown in the center of the applet window. Simultaneously, intensity is shifted away from the central peak of the point spread function and into the surrounding rings, which become far more prominent. These changes are also correlated with the ray trace diagram presented in the right-hand side of the applet.

(a) What diameter mirror is needed to be able to see 1.00 m detail on a Jovian Moon at a distance of 7.50 x 108 km from Earth? The wavelength of light averages 600 nm.

The angle found in part (a) is extraordinarily small (less than 1/50,000 of a degree), because the primary mirror is so large compared with the wavelength of light. As noticed, diffraction effects are most noticeable when light interacts with objects having sizes on the order of the wavelength of light. However, the effect is still there, and there is a diffraction limit to what is observable. The actual resolution of the Hubble Telescope is not quite as good as that found here. As with all instruments, there are other effects, such as non-uniformities in mirrors or aberrations in lenses that further limit resolution. However, Figure 3 gives an indication of the extent of the detail observable with the Hubble because of its size and quality and especially because it is above the Earth’s atmosphere.

Only when the specimen and image distances can be accurately specified can spherical aberration be optimally corrected. This artifact can be easily introduced by improper tube length caused by introduction of optical elements into the converging beam path of finite tube length microscopes. Spherical aberration can also occur when using improper "windows", such as cover glasses of nonstandard thickness (deviations from 0.17 millimeters) or poor quality immersion oil between the objective front lens and the cover glass.

1: The 300-m-diameter Arecibo radio telescope pictured in Figure 4 detects radio waves with a 4.00 cm average wavelength.

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Spherical aberrationimage

Just what is the limit? To answer that question, consider the diffraction pattern for a circular aperture, which has a central maximum that is wider and brighter than the maxima surrounding it (similar to a slit) [see Figure 2(a)]. It can be shown that, for a circular aperture of diameter D, the first minimum in the diffraction pattern occurs at θ = 1.22 λ/D (providing the aperture is large compared with the wavelength of light, which is the case for most optical instruments). The accepted criterion for determining the diffraction limit to resolution based on this angle was developed by Lord Rayleigh in the 19th century. The Rayleigh criterion for the diffraction limit to resolution states that two images are just resolvable when the centre of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. See Figure 2(b). The first minimum is at an angle of  θ = 1.22λ/ D, so that two point objects are just resolvable if they are separated by the angle

All attempts to observe the size and shape of objects are limited by the wavelength of the probe. Even the small wavelength of light prohibits exact precision. When extremely small wavelength probes as with an electron microscope are used, the system is disturbed, still limiting our knowledge, much as making an electrical measurement alters a circuit. Heisenberg’s uncertainty principle asserts that this limit is fundamental and inescapable, as we shall see in quantum mechanics.

Spherical aberration artifacts are encountered when light waves passing through the periphery of an uncorrected convex lens are not brought into focus with those passing through the center. Waves passing near the center of the lens are refracted only slightly, whereas waves passing near the periphery are refracted to a greater degree, producing a variety of different focal points along the optical axis. As a result, the peripheral waves come to a shorter focus (nearer the back of the lens or objective) than do rays traveling through the central or axial area. This is known as longitudinal or axial spherical aberration. Axial aberration is generated by non-spherical wavefronts produced by the objective itself or by improper use of the objective. Some of the most common causes are failure to maintain the designated microscope tube length or the presence of substances between the objective and focal plane having a spurious refractive index.

How does diffraction affect the detail that can be observed when light passes through an aperture? Figure 1(b) shows the diffraction pattern produced by two point light sources that are close to one another. The pattern is similar to that for a single point source, and it is just barely possible to tell that there are two light sources rather than one. If they were closer together, as in Figure 1(c), we could not distinguish them, thus limiting the detail or resolution we can obtain. This limit is an inescapable consequence of the wave nature of light.

8: You are told not to shoot until you see the whites of their eyes. If the eyes are separated by 6.5 cm and the diameter of your pupil is 5.0 mm, at what distance can you resolve the two eyes using light of wavelength 555 nm?

Contrast Limitations for Lenses. Lens contrast is defined as the percentage of contrast on the object that is reproduced into image space when assuming no ...

An amateur astronomer wants to build a telescope with a diffraction limit that will allow him to see if there are people on the moons of Jupiter.

High-quality oil immersion objectives perform optimally only when they are used with a cover glass thickness of 0.17 millimeters. To help alleviate cover glass variations, correction collars are often included on dry objectives to enable adjustment of intermediate lens elements to compensate for deviant cover glass thickness. Because focus may shift and the image may wonder during adjustment of the correction collar, the utilization of correction collars demands that the microscopist remain alert in order to reset the collar using appropriate image criteria. In addition, the insertion of accessories in the light path of finite tube length objectives may introduce aberrations, when the specimen is refocused, unless these accessories have been properly designed with additional optics.

(a) What is the angle between two just-resolvable points of light for a 3.00-mm-diameter pupil, assuming an average wavelength of 550 nm?