Internal reflections, sometimes called "scatter", cause the light passing through an eyepiece to disperse and reduce the contrast of the image projected by the eyepiece. When the effect is particularly bad, "ghost images" are seen, called "ghosting". For many years, simple eyepiece designs with a minimum number of internal air-to-glass surfaces were preferred to avoid this problem.

The generally accepted visual distance of closest focus   D   {\displaystyle \ D\ } is 250 mm, and eyepiece power is normally specified assuming this value. Common eyepiece powers are 8×, 10×, 15×, and 20×. The focal length of the eyepiece (in mm) can thus be determined if required by dividing 250 mm by the eyepiece power.

Beamdivergence

For propagation in transparent media, <$\lambda$> is the wavelength in the medium (i.e., not the vacuum wavelength). Otherwise, the formalism explained above can be used without modification, assuming that the medium is homogeneous, isotropic and lossless.

The modes of an optical resonator with the lowest order in the transverse direction (called TEM00 or fundamental transverse modes) are Gaussian modes, if the resonator is stable, all optical media in the resonator are homogeneous, and all surfaces between media are either flat or have a parabolic shape. Therefore, lasers emitting only on the fundamental transverse mode often emit beams with close to Gaussian shape. Deviations from the mentioned conditions, e.g. by thermal lensing in a gain medium, can cause non-Gaussian beam shapes and/or the simultaneous excitation of different transverse modes.

If I have a pulsed laser with 10 μJ pulse energy and the beam is Gaussian, is that energy the average energy or the peak energy? Would the energy at tip of the distribution (the center of the beam) be equal to 10 μJ or less by a factor?

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The simple negative lens placed before the focus of the objective has the advantage of presenting an erect image but with limited field of view better suited to low magnification. It is suspected this type of lens was used in some of the first refracting telescopes that appeared in the Netherlands in about 1608. It was also used in Galileo Galilei's 1609 telescope design which gave this type of eyepiece arrangement the name "Galilean". This type of eyepiece is still used in very cheap telescopes, binoculars and in opera glasses.

The eye needs to be held at a certain distance behind the eye lens of an eyepiece to see images properly through it. This distance is called the eye relief. A larger eye relief means that the optimum position is farther from the eyepiece, making it easier to view an image. However, if the eye relief is too large it can be uncomfortable to hold the eye in the correct position for an extended period of time, for which reason some eyepieces with long eye relief have cups behind the eye lens to aid the observer in maintaining the correct observing position. The eye pupil should coincide with the exit pupil, the image of the entrance pupil, which in the case of an astronomical telescope corresponds to the object glass.

In terms of Gaussian beam parameters, the paraxial approximation requires that the beam radius at the focus is large compared with the wavelength. (However, it does not need to be far larger for reasonable accuracy of the equations.) This implies that the beam divergence does not become too large, and that the Rayleigh length is substantially larger than the beam radius. For very tightly focused beams, the paraxial approximation is not well satisfied, and a substantially more complex method is required for calculating the beam propagation. In that regime, one may also need to consider the vector character of the electromagnetic field, including a longitudinal polarization component.

The above formulas are approximations. The ISO 14132-1:2002 standard gives the exact calculation for apparent field of view,   A F O V   , {\displaystyle \ A_{\mathsf {FOV}}\ ,} from the true field of view,   T F O V   , {\displaystyle \ T_{\mathsf {FOV}}\ ,} as:

In addition to the Gaussian shape of the intensity profile, a Gaussian beam has a transverse phase profile which can be described with a polynomial of at most second order:

The Plössl is an eyepiece usually consisting of two sets of doublets, designed by Georg Plössl in 1860. Since the two doublets can be identical this design is sometimes called a symmetrical eyepiece.[6] The compound Plössl lens provides a large 50° or more apparent field of view, along with the proportionally large true FOV. This makes this eyepiece ideal for a variety of observational purposes including deep-sky and planetary viewing. The chief disadvantage of the Plössl optical design is short eye relief compared to an orthoscopic, since the Plössl eye relief is restricted to about 70–80% of focal length. The short eye relief is more critical in short focal lengths below about 10 mm, when viewing can become uncomfortable – especially for people wearing glasses.

Since   M =   f T   f E   , {\displaystyle \ M={\frac {\ f_{\mathsf {T}}\ }{f_{\mathsf {E}}}}\ ,} where:

A separation of exactly 1 focal length is also inadvisable since it renders the dust on the field lens disturbingly in focus. The two curved surfaces face inwards. The focal plane is thus located outside of the eyepiece and is hence accessible as a location where a graticule, or micrometer crosshairs may be placed. Because a separation of exactly one focal length would be required to correct transverse chromatic aberration, it is not possible to correct the Ramsden design completely for transverse chromatic aberration. The design is slightly better than Huygens but still not up to today's standards.

If the apparent field of view is known, the actual field of view can be calculated from the following approximate formula:

where   f A   {\displaystyle \ f_{\mathsf {A}}\ } and   f B   {\displaystyle \ f_{\mathsf {B}}\ } are the focal lengths of the component lenses.

Definition: light beams where the electric field profile in a plane perpendicular to the beam axis can be described with a Gaussian function, possibly with an added parabolic phase profile

Laguerre Gaussbeam

Several properties of an eyepiece are likely to be of interest to a user of an optical instrument, when comparing eyepieces and deciding which eyepiece suits their needs.

Elements are the individual lenses, which may come as simple lenses or "singlets" and cemented doublets or (rarely) triplets. When lenses are cemented together in pairs or triples, the combined elements are called groups (of lenses).

Gaussian beams are usually (and also in this article) considered in situations where the beam divergence is relatively small (i.e., the beam waist radius sufficiently large), so that the so-called paraxial approximation can be applied. This approximation allows the omission of the term with the second-order derivative in the propagation equation (as derived from Maxwell's equations), so that a first-order differential equation results. Within the paraxial approximation, a Gaussian beam propagating in free space (or in a homogeneous medium) remains Gaussian, except that of course its parameters evolve. For a monochromatic beam, propagating in the <$z$> direction with the wavelength <$\lambda$>, the complex electric field amplitude (phasor) is

An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is named because it is usually the lens that is closest to the eye when someone looks through an optical device to observe an object or sample. The objective lens or mirror collects light from an object or sample and brings it to focus creating an image of the object. The eyepiece is placed near the focal point of the objective to magnify this image to the eyes. (The eyepiece and the eye together make an image of the image created by the objective, on the retina of the eye.) The amount of magnification depends on the focal length of the eyepiece.

It is common for users of an eyepiece to want to calculate the actual field of view, because it indicates how much of the sky will be visible when the eyepiece is used with their telescope. The most convenient method of calculating the actual field of view depends on whether the apparent field of view is known.

They obviously use a two times smaller value of Io, which is then only half the peak intensity. I am not sure why one would like to do that.

It seems that you got the integration wrong: you apparently simply integrated the Gaussian function in the radial direction from 0 to <$w$>. However, you need to integrate over the area; therefore, the integrand must be <$\exp(-2 (r / w)^2) \: 2\pi \: r \: \textrm{d}r$> (apart from the normalization factor). With that, you get the mentioned 86.5%.

(With electrical engineer's sign convention, the sign of the term with <$\lambda / \pi w^2$> in the definition of <$q(z)$> would be opposite. Effectively, <$q$> is turned into its complex conjugate.)

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Due to the basic phenomenon of diffraction, the beam radius cannot simply remain constant – it varies along the propagation direction. This can be described mathematically as

with the peak amplitude <$|E_0|$> and beam radius <$w_0$> at the beam waist, the wavenumber <$k = 2\pi / \lambda$>, the Rayleigh length <$z_\textrm{R}$> (see below) and the radius of curvature <$R(z)$> of the wavefronts. The oscillating real electric field is obtained by multiplying the phasor with <$\exp(-i \omega t) = \exp(-i 2\pi c t / \lambda)$> and taking the real part.

Technology has developed over time and there are a variety of eyepiece designs for use with telescopes, microscopes, gun-sights, and other devices. Some of these designs are described in more detail below.

The focal length of an eyepiece is the distance from the principal plane of the eyepiece to where parallel rays of light converge to a single point. When in use, the focal length of an eyepiece, combined with the focal length of the telescope or microscope objective, to which it is attached, determines the magnification. It is usually expressed in millimetres when referring to the eyepiece alone. When interchanging a set of eyepieces on a single instrument, however, some users prefer to identify each eyepiece by the magnification produced.

One solution to scatter is to use thin film coatings over the surface of the element. These thin coatings are only one or two wavelengths deep, and work to reduce reflections and scattering by changing the refraction of the light passing through the element. Some coatings may also absorb light that is not being passed through the lens in a process called total internal reflection where the light incident on the film is at a shallow angle.

If I want to calculate the radius of a Gaussian beam using a knife edge between thresholds, say 10% - 90%, or 20% - 80%, or 25% - 75%, how do I do it? I have seen in scientific literature that for a threshold of 10% - 90%, the radius, r, of the beam r = 0.7803 · (t2 - t1), where t2, and t1 are distances at 90% and 10% thresholds. They did not show how it is done.

For me a 86.5% level of the normal distribution would correspond to 1.5 standard deviations. Increasing the beam waist by 2 would mean 3 standard deviations and rather 99.7% than 99.97% intensity.

The number of elements in a Nagler makes them seem complex, but the idea of the design is fairly simple: every Nagler has a negative doublet field lens, which increases magnification, followed by several positive groups. The positive groups, considered separate from the first negative group, combine to have long focal length, and form a positive lens. That allows the design to take advantage of the many good qualities of low power lenses. In effect, a Nagler is a superior version of a Barlow lens combined with a long focal length eyepiece. This design has been widely copied in other wide field or long eye relief eyepieces.

It has a horizontal field of view of 90 degrees. If you take an image and view it on your phone at a distance of 50cm it will look really ...

An RKE eyepiece has an achromatic field lens and double convex eye lens, a reversed adaptation of the Kellner eyepiece, with its lens layout similar to the König. It was designed by Dr. David Rank for the Edmund Scientific Corporation, who marketed it throughout the late 1960s and early 1970s. This design provides slightly wider field of view than classic Kellner design and makes its design similar to a widely spaced version of the König.

Modern instruments often use objectives optically corrected for an infinite tube length rather than 160 mm, and these require an auxiliary correction lens in the tube.

In some eyepiece types, such as Ramsden eyepieces (described in more detail below), the eyepiece behaves as a magnifier, and its focal plane is located outside of the eyepiece in front of the field lens. This plane is therefore accessible as a location for a graticule or micrometer crosswires. In the Huygenian eyepiece, the focal plane is located between the eye and field lenses, inside the eyepiece, and is hence not accessible.

Pulse energy is always the energy over the full beam area. A local quantity would be the pulse fluence = energy per area in units of J/cm2, for example. That thing integrated over the full beam area is the pulse energy.

The Ramsden eyepiece comprises two plano-convex lenses of the same glass and similar focal lengths, placed less than one eye-lens focal length apart, a design created by astronomical and scientific instrument maker Jesse Ramsden in 1782. The lens separation varies between different designs, but is typically somewhere between ⁠ 7 /10⁠ and ⁠ 7 /8⁠ of the focal length of the eye-lens, the choice being a trade off between residual transverse chromatic aberration (at low values) and at high values running the risk of the field lens touching the focal plane when used by an observer who works with a close virtual image such as a myopic observer, or a young person whose accommodation is able to cope with a close virtual image (this is a serious problem when used with a micrometer as it can result in damage to the instrument).

The total angular magnification of a microscope image is then simply calculated by multiplying the eyepiece power by the objective power. For example, a 10× eyepiece with a 40× objective will magnify the image 400 times.

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Modern improvements typically have fields of view of 60°−70°. König design revisions use exotic glass and / or add more lens groups; the most typical adaptation is to add a simple positive, concave-convex lens before the doublet, with the concave face towards the light source and the convex surface facing the doublet.

A simple convex lens placed after the focus of the objective lens presents the viewer with a magnified inverted image. This configuration may have been used in the first refracting telescopes from the Netherlands and was proposed as a way to have a much wider field of view and higher magnification in telescopes in Johannes Kepler's 1611 book Dioptrice. Since the lens is placed after the focal plane of the objective it also allowed for use of a micrometer at the focal plane (used for determining the angular size and/or distance between objects observed).

By convention, microscope eyepieces are usually specified by power instead of focal length. Microscope eyepiece power   P E   {\displaystyle \ P_{\mathrm {E} }\ } and objective power   P O   {\displaystyle \ P_{\mathsf {O}}\ } are defined by

Propagation over some distance then simply increases the <$q$> parameter by that distance. When a Gaussian beam passes an optical element such as a curved mirror or a lens, this can be described by transforming its parameters with an ABCD matrix according to

The eyepieces of binoculars are usually permanently mounted in the binoculars, causing them to have a pre-determined magnification and field of view. With telescopes and microscopes, however, eyepieces are usually interchangeable. By switching the eyepiece, the user can adjust what is viewed. For instance, eyepieces will often be interchanged to increase or decrease the magnification of a telescope. Eyepieces also offer varying fields of view, and differing degrees of eye relief for the person who looks through them.

In a Kellner eyepiece an achromatic doublet is used in place of the simple plano-convex eye lens in the Ramsden design to correct the residual transverse chromatic aberration. Carl Kellner designed this first modern achromatic eyepiece in 1849,[4] also called an "achromatized Ramsden". Kellner eyepieces are a 3-lens design. They are inexpensive and have fairly good image from low to medium power and are far superior to Huygenian or Ramsden design. The eye relief is better than the Huygenian and worse than the Ramsden eyepieces.[5] The biggest problem of Kellner eyepieces was internal reflections. Today's anti-reflection coatings make these usable, economical choices for small to medium aperture telescopes with focal ratio f/6 or longer. The typical apparent field of view is 40–50°.

Eyepieces are optical systems where the entrance pupil is invariably located outside of the system. They must be designed for optimal performance for a specific distance to this entrance pupil (i.e. with minimum aberrations for this distance). In a refracting astronomical telescope the entrance pupil is identical with the objective. This may be several feet distant from the eyepiece; whereas with a microscope eyepiece the entrance pupil is close to the back focal plane of the objective, mere inches from the eyepiece. Microscope eyepieces may be corrected differently from telescope eyepieces; however, most are also suitable for telescope use.

Note that for <$z \rightarrow 0$> one obtains <$R \rightarrow \infty$> rather than <$R \rightarrow 0$>, as we have <$z^2$> in the denominator.

Consider a plane wave propagating with an angle <$\varphi$> against the z direction. The wave vector has an x component, for example, with the magnitude <$k \: \sin\varphi$>. That component describes a linear phase variation in x direction.

According to Edmund Scientific Corporation, RKE stands for "Rank Kellner Eyepiece'".[citation needed] In an amendment to their trademark application on 16 January 1979 it was given as "Rank-Kaspereit-Erfle", the three designs from which the eyepiece was derived.[15] Edmund Astronomy News (March 1978) called the eyepiece the "Rank-Kaspereit-Erfle" (RKE) a "redesign[ed] ... type II Kellner".[16] However, the RKE deign does not resemble a Kellner, and is closer to a modified König. There is some speculation that at some point the "K" was mistakenly interpreted as the name of the more common Kellner, instead of the fairly rarely seen König.

It is often defined like that, but one may use a generalized definition where you require Gaussians in <$x$> and <$y$> direction, but not necessarily with the same width.

The main disadvantage to Naglers is in their weight; they are often ruefully referred to as ‘hand grenades’ because of their heft and large size. Long focal length versions exceed 0.5 kg (1.1 lb), which is enough to unbalance small to medium-sized telescopes. Another disadvantage is a high purchase cost, with large Naglers' prices comparable to the cost of a small telescope. Hence these eyepieces are regarded by many amateur astronomers as a luxury.[19]

Huygens eyepieces consist of two plano-convex lenses with the plane sides towards the eye separated by an air gap. The lenses are called the eye lens and the field lens. The focal plane is located between the two lenses. It was invented by Christiaan Huygens in the late 1660s and was the first compound (multi-lens) eyepiece.[2] Huygens discovered that two air spaced lenses can be used to make an eyepiece with zero transverse chromatic aberration. If the lenses are made of glass of the same Abbe number, to be used with a relaxed eye and a telescope with an infinitely distant objective then the separation is given by:

This formula also indicates that, for an eyepiece design with a given apparent field of view, the barrel diameter will determine the maximum focal length possible for that eyepiece, as no field stop can be larger than the barrel itself. For example, a Plössl with 45° apparent field of view in a 1.25 inch barrel would yield a maximum focal length of 35 mm.[1] Anything longer requires larger barrel or the view is restricted by the edge, effectively making the field of view less than 45°.

An Erfle is a 5 element eyepiece consisting of 2 achromatic doublets with an extra simple lens between them. They were invented by Heinrich Erfle during World War I for military use.[14] The design is an elementary extension of 4 element eyepieces such as Plössls, enhanced for wider fields.

Radians (or mrad) are not a real unit, as the radian measure is essentially a ratio (of circumference length to radius), thus a dimensionless ratio. We use the radians only to indicate that we use that radian concept to specify an angle. So the basic units on each side of the equation are meters.

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This definition of lens power relies upon an arbitrary decision to split the angular magnification of the instrument into separate factors for the eyepiece and the objective. Historically, Abbe described microscope eyepieces differently, in terms of angular magnification of the eyepiece and 'initial magnification' of the objective. While convenient for the optical designer, this turned out to be less convenient from the viewpoint of practical microscopy and was thus subsequently abandoned.

where <$A$>, <$B$>, <$C$> and <$D$> are the components of the ABCD matrix. In textbooks, ABCD matrices for many kinds of optical elements are available. Note that with physicists' sign convention, all matrix components need to be turned into their complex conjugates; however, in many cases of interest, the matrix components are purely real.

Magnification increases, therefore, when the focal length of the eyepiece is shorter or the focal length of the objective is longer. For example, a 25 mm eyepiece in a telescope with a 1200 mm focal length would magnify objects 48 times. A 4 mm eyepiece in the same telescope would magnify 300 times.

The second formula is actually more accurate, but field stop size is not usually specified by most manufacturers. The first formula will not be accurate if the field is not flat, or is higher than 60° which is common for most ultra-wide eyepiece design.

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Calculate the effect of a half wave plate at an arbitrary θ on horizontally polarized light. Solution: Multiplying by the half-wave matrix (6.39), we obtain.

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The article on laser beams contains a paragraph titled “Limitations for the Focusing of Laser Beams”. The presented rules can be applied to Gaussian beams but also to generalize beams with some larger M2 factor.

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These eyepieces work well with the very long focal length telescopes.[c] This optical design is now considered obsolete since with today's shorter focal length telescopes the eyepiece suffers from short eye relief, high image distortion, axial chromatic aberration, and a very narrow apparent field of view. Since these eyepieces are cheap to make they can often be found on inexpensive telescopes and microscopes.[3]

Laser cutting

Note that there are also multimode beams with Gaussian intensity profile but complicated phase patterns, and these are not called Gaussian beams.

If a diagonal or Barlow lens is used before the eyepiece, the eyepiece's field of view may be slightly restricted. This occurs when the preceding lens has a narrower field stop than the eyepiece's, causing the obstruction in the front to act as a smaller field stop in front of the eyepiece. The exact relationship is given by

Amateur astronomers tend to refer to telescope eyepieces by their focal length in millimeters. These typically range from about 3 mm to 50 mm. Some astronomers, however, prefer to specify the resulting magnification power rather than the focal length. It is often more convenient to express magnification in observation reports, as it gives a more immediate impression of what view the observer actually saw. Due to its dependence on properties of the particular telescope in use, however, magnification power alone is meaningless for describing a telescope eyepiece.

You just need to integrate the intensity of the Gaussian beam over an area which is cut off to some extent on one side according to the area covered by the knife edge. That can be done with an analytical calculation or with numerical tools, as is in both cases not particularly difficult.

Erfle eyepieces are designed to have wide field of view (about 60°), but are unusable at high powers because they suffer from astigmatism and ghost images.[d] However, with lens coatings at low powers (focal lengths of 20~30 mm and up) they are acceptable, and at 40 mm they can be excellent. Erfles are very popular for wide-field views, because they have large eye lenses, and can be very comfortable to use because of their good eye relief in longer focal lengths.

Simply integrate the Gaussian beam intensity over the area of a circle with the given radius, and divided that by the full area integral. Use polar coordinates to obtain:

which shows that the smaller the waist radius and the longer the wavelength, the stronger is the divergence of the beam far from the waist. The beam parameter product (product of waist radius and far-field divergence angle) of a Gaussian beam is <$\lambda /\pi$>, i.e., it depends only the wavelength. For light beams with non-ideal beam quality (see below), that value is larger.

You find that value if you calculate the integral over the Gaussian function, e.g. with a peak value of 1 and a given FWHM, in order to normalize it to a certain energy.

There are no requirements concerning polarization of a Gaussian beam, i.e., it may e.g. be linear, circular, elliptical, radial, azimuthal, or not polarized at all. In many cases, a scalar description of a Gaussian beam, ignoring polarization, is used.

(Note: the equation above is based on the physicists' sign convention for wave phasors, rather than the one which is more common in engineering, although in the optics one often finds the latter in the scientific literature, where the last phase term is <$\exp(-j \: [...])$>. If that is used, the signs in some other equations need to be adapted. See the article on sign conventions in wave optics for more details.)

Beamquality

In any case, the deviation from a Gaussian beam shape can be quantified with the M2 factor. A Gaussian beam has the highest possible beam quality, which is related to the lowest possible beam parameter product, and corresponds to <$M^2$> = 1.

Gaussian beams can have different radii and divergence values for two perpendicular transverse directions, denoted e.g. <$x$> and <$y$>. Equations similar to those above can be used for describing the essentially independent evolution of beam radii in both directions. If the focus positions for both directions are not equal, the beam is called astigmatic.

The transverse profile of the optical intensity of the beam with an optical power <$P$> can be described with a Gaussian function:

In optics and particularly in laser physics, laser beams often occur in the form of Gaussian beams, which are named after the mathematician and physicist Johann Carl Friedrich Gauß. The definition of Gaussian beams concerns both the intensity and phase profile, as explained in the following:

Gaussianbeam

The term <$-\arctan z/z_\textrm{R}$> in the expression for the phase of the electric field describes the Gouy phase shift, which is important e.g. for the resonance frequencies of optical resonators.

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Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

Lateral or transverse chromatic aberration is caused because the refraction at glass surfaces differs for light of different wavelengths. Blue light, seen through an eyepiece element, will not focus to the same point but along the same axis as red light. The effect can create a ring of false colour around point sources of light and results in a general blurriness to the image.

The position <$z = 0$> in the equation above corresponds to the beam waist or focus where the beam radius is at its minimum, and the phase profile is flat. The radius of curvature <$R$> of the wavefronts evolves according to

The focal length of the telescope objective,   f T   , {\displaystyle \ f_{\mathsf {T}}\ ,} is the diameter of the objective times the focal ratio. It represents the distance at which the mirror or objective lens will cause light from a star to converge onto a single point (aberrations excepted).

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I would expect the power transmission of a Gaussian beam through a 1-w radius aperture to be the integral of your expression for the intensity, which I'm calculating as 95.4%, not 86.5%.

The first eyepieces had only a single lens element, which delivered highly distorted images. Two and three-element designs were invented soon after, and quickly became standard due to the improved image quality. Today, engineers assisted by computer-aided drafting software have designed eyepieces with seven or eight elements that deliver exceptionally large, sharp views.

The 4-element orthoscopic eyepiece consists of a plano-convex singlet eye lens and a cemented convex-convex triplet field lens achromatic field lens. This gives the eyepiece a nearly perfect image quality and good eye relief, but a narrow apparent field of view — about 40°–45°. It was invented by Ernst Abbe in 1880.[3] It is called "orthoscopic" or "orthographic" because of its low degree of distortion and is also sometimes called an "ortho" or "Abbe".

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Eyepieces for telescopes and microscopes are usually interchanged to increase or decrease the magnification, and to enable the user to select a type with certain performance characteristics. To allow this, eyepieces come in standardized "Barrel diameters".

What happens when a Gaussian beam goes into another medium, passing by a planar interface? Is it just a matter of correcting parameters like beam divergence, Rayleigh range and Gouy phase by considering the new refractive index and correcting the beam direction using Snell's law?

which determines the length over which the beam can propagate without diverging significantly. (The older literature often deals with the confocal length <$b$>, which is just twice the Rayleigh length.) A so-called collimated beam (with approximately constant beam radius) has to have a large Rayleigh length, compared with the envisaged propagation distance.

The fundamental propagation modes of fibers are generally not exactly Gaussian, but also not too far from that shape. Therefore, a Gaussian beam can usually be launched into a single-mode fiber with high efficiency (80% or larger), provided that suitable optics are used. One requires a focus at the fiber end with a beam radius which fits to the size of the fiber mode.

An eyepiece consists of several "lens elements" in a housing, with a "barrel" on one end. The barrel is shaped to fit in a special opening of the instrument to which it is attached. The image can be focused by moving the eyepiece nearer and further from the objective. Most instruments have a focusing mechanism to allow movement of the shaft in which the eyepiece is mounted, without needing to manipulate the eyepiece directly.

There are six standard barrel diameters for telescopes. The barrel sizes (usually expressed in inches[citation needed]) are:

A Monocentric is an achromatic triplet lens with two pieces of crown glass cemented on both sides of a flint glass element. The elements are thick, strongly curved, and their surfaces have a common center giving it the name "monocentric". It was invented by H.A. Steinheil around 1883.[9] This design, like the solid eyepiece designs of Tolles, Hastings, and Taylor,[10] is free from ghost reflections and gives a bright contrasty image, a desirable feature when it was invented (before anti-reflective coatings).[11] It has a narrow apparent field of view around 25°[12] but was favored by planetary observers.[13]

Please help me identify the approach to finding the beam waist radius after it has undergone focusing by a lens of a given focal length, given the initial radius (before focusing) and its wavelength.

One solution is to reduce the aberration by using multiple elements of different types of glass. Achromats are lens groups that bring two different wavelengths of light to the same focus and exhibit greatly reduced false colour. Low dispersion glass may also be used to reduce chromatic aberration.

The field of view, often abbreviated FOV, describes the area of a target (measured as an angle from the location of viewing) that can be seen when looking through an eyepiece. The field of view seen through an eyepiece varies, depending on the magnification achieved when connected to a particular telescope or microscope, and also on properties of the eyepiece itself. Eyepieces are differentiated by their field stop, which is the narrowest aperture that light entering the eyepiece must pass through to reach the field lens of the eyepiece.

Until the advent of multicoatings and the popularity of the Plössl, orthoscopics were the most popular design for telescope eyepieces. Even today these eyepieces are considered good eyepieces for planetary and lunar viewing. They are preferred for reticle eyepieces, since they are one of the wide-field, long eye-relief designs with an external focal plane; slowly being supplanted by the König. Due to their low degree of distortion and the corresponding globe effect, they are less suitable for applications which require an extensive panning of the instrument.

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There are some optical elements which can not be described with ABCD matrices, and which convert a Gaussian beam into a non-Gaussian beam; an example are axicons.

The Plössl eyepiece was an obscure design until the 1980s when astronomical equipment manufacturers started selling redesigned versions of it.[7] Today it is a very popular design on the amateur astronomical market,[8] where the name Plössl covers a range of eyepieces with at least four optical elements, sometimes overlapping with the Erfle design.

This eyepiece is one of the more expensive to manufacture because of the quality of glass, and the need for well matched convex and concave lenses to prevent internal reflections. Due to this fact, the quality of different Plössl eyepieces varies. There are notable differences between cheap Plössls with simplest anti-reflection coatings and well made ones.

The König eyepiece has a concave-convex positive doublet and a plano-convex singlet. The strongly convex surfaces of the doublet and singlet face and (nearly) touch each other. The doublet has its concave surface facing the light source and the singlet has its almost flat (slightly convex) surface facing the eye. It was designed in 1915 by German optician Albert König (1871−1946)[citation needed] and is effectively a simplified Abbe. The design allows for high magnification with remarkably high eye relief – the longest eye relief proportional to focal length of any design before the Nagler, in 1979. The field of view of about 55° is slightly superior to the Plössl, with the further advantages of better eye relief and requiring one less lens element.

Laser M2

Here, the beam radius <$w(z)$> is the distance from the beam axis where the intensity drops to <$1/e^2$> (≈ 13.5%) of the maximum value. If the beam hits a hard aperture with radius <$w$>, ≈ 86.5% of the optical power can get through the aperture. For an aperture radius of 1.5 <$w$> or 2 <$w$>, this fraction is increased to 98.9% and 99.97%, respectively. (A common error in the integration leads to substantially different results – see section “Questions and Comments from Users” below.)

Because Huygens eyepieces do not contain cement to hold the lens elements, telescope users sometimes use these eyepieces in the role of "solar projection", i.e. projecting an image of the Sun onto a screen for prolonged periods of time. Cemented eyepieces are traditionally regarded as potentially vulnerable to heat damage by the intense concentrations of light involved.

For a telescope, the approximate angular magnification   M A   {\displaystyle \ M_{\mathsf {A}}\ } produced by the combination of a particular eyepiece and objective can be calculated with the following formula:

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Eye relief typically ranges from about 2 mm to 20 mm, depending on the construction of the eyepiece. Long focal-length eyepieces usually have ample eye relief, but short focal-length eyepieces are more problematic. Until recently, and still quite commonly, eyepieces of a short-focal length have had a short eye relief. Good design guidelines suggest a minimum of 5–6 mm to accommodate the eyelashes of the observer to avoid discomfort. Modern designs with many lens elements, however, can correct for this, and viewing at high power becomes more comfortable. This is especially the case for spectacle wearers, who may need up to 20 mm of eye relief to accommodate their glasses.

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Invented by Albert Nagler and patented in 1979, the Nagler eyepiece is a design optimized for astronomical telescopes to give an ultra-wide field of view (82°) that has good correction for astigmatism and other aberrations. Introduced in 2007, the Ethos is an enhanced ultra-wide field design developed principally by Paul Dellechiaie under Albert Nagler's guidance at Tele Vue Optics and claims a 100–110° AFOV.[17][18] This is achieved using exotic high-index glass and up to eight optical elements in four or five groups; there are several similar designs called the Nagler, Nagler type 2, Nagler type 4, Nagler type 5, and Nagler type 6. The newer Delos design is a modified Ethos design with a FOV of 'only' 72 degrees but with a long 20 mm eye relief.

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Note that the factor 1 / 2 in the denominator in the equation is unfortunately often forgotten, so that the on-axis intensity of the beam is underestimated by a factor of 2. For example, quoted numbers for the measured damage threshold of optical components are often affected by that problem; the peak intensity at the damage threshold in terms of optical power may have be calculated with or without the mentioned factor, so that a substantial quantitative uncertainty remains for the reader.

Longitudinal chromatic aberration is a pronounced effect of optical telescope objectives, because the focal lengths are so long. Microscopes, whose focal lengths are generally shorter, do not tend to suffer from this effect.