Physics
Spherical Aberration
Spherical aberration is a type of optical distortion that occurs in spherical lenses or mirrors, causing light rays to focus at different points. This results in a blurred or distorted image. It is caused by the spherical shape of the lens or mirror, which leads to varying focal lengths for different parts of the lens.
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12 Key excerpts on "Spherical Aberration"
eBook - PDFLens Design
A Practical Guide
- Haiyin Sun(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
55 2 Optical Aberrations Any real optical system contains various aberrations. The main task of optical design is to minimize these aberrations. In this chapter, we briefly describe the five important aberrations: Spherical Aberration, coma, astigmatism, field curvature, and image distortion. These five aberrations are monochromatic. We will also describe two chromatic aberrations: longitudinal color and lateral color. 2.1 Spherical Aberration 2.1.1 REFLECTION Spherical Aberration The most commonly seen aberration is the Spherical Aberration. Figure 2.1a shows a spherical mirror focusing a ray parallel to its optical axis. The mirror surface has a radius of curvature R, the incident ray has a height of h to the optical axis, and the focused spot is a distance of x away from the center of surface curvature. The incident ray hits one point on the mirror surface and has an angle of θ to the normal of this point on the surface. The reflected ray also has an angle of θ to the normal according to the reflection law. From Figure 2.1a, we have sin( ) θ = h R (2.1) cos( ) θ = R x /2 (2.2) Equation 2.2 holds because the distance between the focused spot and the point where the ray hits the mirror surface also equals to x. Combining Equations 2.1 and 2.2 to solve for the Spherical Aberration SA = x − R/2, we obtain SA x R R h R = - = - - 2 2 1 1 1 2 0 5 . (2.3) From Equation 2.3, we can see that for a given mirror, the Spherical Aberration varies as the height of the incident ray varies. When h → 0, SA → 0 and x → R/2. So, R/2 is the paraxial focal length of the mirror. We usually omit the term “par- axial” and simply use the term “focal length.” For h > 0, SA > 0. 56 Lens Design Plotted in Figure 2.1b is a raytracing diagram generated by Zemax showing the Spherical Aberration of a spherical mirror. Spherical Aberration prevents a group of parallel rays being focused at the same spot and is an undesired property.
eBook - PDFOptical Imaging and Photography
Introduction to Science and Technology of Optics, Sensors and Systems
- Ulrich Teubner, Hans Josef Brückner(Authors)
- 2019(Publication Date)
- De Gruyter(Publisher)
Putting an imaginary screen in vertical direction to the center of their paraxial focal points, the images could be characterized as a spot in the center surrounded by a circular halo. 166 | 3 Imaging optics (b) (a) Fig. 3.39: Spherical Aberration and caustic in different situations. (a) Refraction of sunlight at a drinking glass filled with water acting as a cylindrical lens showing spherical and chromatic aberra-tions; (b) reflection of sunlight at a cylindrical surface. A similar image is expected on a screen in the focal plane in Figure 3.38a. It should be noted that in this figure the additional chromatic aberration becomes also visible with the outer edge of the caustic envelope having a red color. This combination of spher-ical and chromatic aberration is termed spherochromatism and will be discussed in the section of chromatic aberrations. As a consequence the Spherical Aberration in an image deteriorates its quality with respect to sharpness as well as image contrast. As stated above, the mathematical treatment of the Spherical Aberration is done only for object points on the optical axis. For off-axis points additional phenomena as described in the next sections show up. The third order theory predicts a longitudinal Spherical Aberration with z s ∝ h 2 as well as the transversal aberration being propor-tional to h 3 . The parabolic relationship of z s is indicated in a diagram in Figure 3.38. It becomes evident that reducing the aperture of an incident light bundle, for instance by an aperture stop, leads to drastic reduction of both longitudinal and transversal aberrations. The position of the stop, however, is uncritical and does not influence them. Furthermore, when the aperture is stopped down, the circle of least confusion is shifted towards the paraxial focal plane besides reducing its spot diameter. The Spherical Aberration strongly depends on the conditions under which the imaging takes place, for instance the object distance and the type of lenses.
- Bernd Jahne(Author)
- 2004(Publication Date)
- CRC Press(Publisher)
All these aberrations have their common cause in the refraction and reflection law. First-order optics, which is the basis of the simple image equation, requires paraxial rays which have a low inclination α and offset to the optical axis so that tan α ≈ sin α ≈ α. (4.30) Only in this limit, mirrors and lenses with spherical shape focus a parallel light beam onto a single point. Additionally, so-called chromatic aberrations are caused by the wavelength dependency of the index of refraction (Section 3.3.1a). For larger angles α and object height h , we can expect aberrations. Seven primary aberrations can be distinguished which are summarized in Table 4.1 and Fig. 4.13. Spherical Aberration is related to the focusing of an axial parallel bundle of monochro-matic light. Each radial zone of the lens aperture has a slightly different focal length. For a simple planoconvex lens, the focal length decreases with the distance of the ray from the optical axis. The lack of a common focus results in a certain spot size onto which a parallel light bundle is focused. Due to the larger area of the off-axis rays, the minimum blur circle is found slightly short off the focus for paraxial rays. 4.3 Concepts 135 astigmatism field curvature Spherical Aberration coma & distortion Figure 4.13: Schematic illustration of the five primary monochromatic aberrations: Spherical Aberration, coma, astigmatism, field curvature, and distortion. Coma is an aberration that affects only off-axis light bundles. For such a light bundle, even rays piercing the lens aperture at a radial zone are not focused onto a point but — because of the lack of symmetry — onto a blur circle with a spot size increasing with the radius of the zone. Also, in contrast to Spherical Aberration, each radial zone focuses onto the image plane at a slightly different height. The result is a spot of comatic shape with a bright central core and a triangular shaped flare extending toward the optical axis.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Optical Aberration Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after trans-mission through the system does not converge into (or does not diverge from) a single point. Instrument-makers need to correct optical systems to compensate for aberration. The articles on reflection, refraction and caustics discuss the general features of reflected and refracted rays. Overview Aberrations fall into two classes: monochromatic and chromatic . Monochromatic aber-rations are caused by the geometry of the lens and occur both when light is reflected and when it is refracted. They appear even when using monochromatic light, hence the name. Chromatic aberrations are caused by dispersion, the variation of a lens's refractive index with wavelength. They do not appear when monochromatic light is used. Monochromatic aberration The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space. The introduction of simple auxiliary terms, due to C. F. Gauss ( Dioptrische Untersuchungen , Göttingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system. The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e. with infinitesimal objects, images and lenses; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.
eBook - PDF- Yen, Anthony, Yu, Shinn-Sheng(Authors)
- 2018(Publication Date)
7 Aberrations in Optical Imaging Systems In this chapter, we investigate image imperfection, which results mainly from the phase errors in image‐forming waves. The phase error, especially in the exit pupil plane, is conventionally called aberration. (Apodization is the word for describing the corresponding amplitude error.) Aberration is generally expressed in polynomials. We first focus on the aberration of optical systems with axial symmetry and study the effects of Seidel aberrations, the lowest‐order aberrations that degrade the image contrast. We then consider the aberration of general optical systems by introducing Zernike polynomials. The orthogonality of Zernike polynomials sheds light on the method of minimizing the aberration when optimizing an optical system. 7.1 Design of Aspherical Surfaces The surfaces of the optical elements in an optical system are spherical in general for easy manufacturing. Such a system, however, cannot produce a spherical wavefront, which is required to produce a stigmatic, or sharp, image. This has nothing to do with manufacturing errors but is related to the intrinsic properties of the optical design. Aberration measures the deviation from a spherical wavefront. In this section, we first present how to reduce aberration by introducing aspherical surfaces. Specifically, using one aspherical surface, one can achieve axial stigmatism, and using two aspherical surfaces, one can achieve aplanatism, i.e., not only axial stigmatism but also the satisfaction of the sine condition, a prerequisite for optical systems in lithographic imaging. In other words, aplanatism is the achievement of sharp images for on‐axis as well as off‐axis object points. How off‐axis can it be? To the extent that the sine condition remains valid! 7.1.1 Attainment of axial stigmatism using one aspherical surface I] As shown in Fig. 7.1‐1(upper), the image ܲ ᇱ (of an on‐axis object ܲ ) formed by the original optical I] B&W, §4.10, and references therein 291
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
____________________ WORLD TECHNOLOGIES ____________________ Chapter 4 Optical Aberration Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge into (or does not diverge from) a single point. Instrument-makers need to correct optical systems to compensate for aberration. The articles on reflection, refraction and caustics discuss the general features of reflected and refracted rays. Overview Aberrations fall into two classes: monochromatic and chromatic . Monochromatic aberrations are caused by the geometry of the lens and occur both when light is reflected and when it is refracted. They appear even when using monochromatic light, hence the name. Chromatic aberrations are caused by dispersion, the variation of a lens's refractive index with wavelength. They do not appear when monochromatic light is used. Monochromatic aberrations • Piston • Tilt • Defocus • Spherical Aberration • Coma • Astigmatism • Field curvature • Image distortion Piston and tilt are not actually true optical aberrations, as they do not represent or model curvature in the wavefront. If an otherwise perfect wavefront is aberrated by piston and ____________________ WORLD TECHNOLOGIES ____________________ tilt, it will still form a perfect, aberration-free image, only shifted to a different position. Defocus is the lowest-order true optical aberration. Chromatic aberrations • Axial, or longitudinal, chromatic aberration • Lateral, or transverse, chromatic aberration Monochromatic aberration The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space. The introduction of simple auxiliary terms, due to C.
eBook - PDF- D.W.O. Heddle(Author)
- 2000(Publication Date)
- CRC Press(Publisher)
5 Aberrations So far we have considered only the focusing of paraxial rays of charged particles homogeneous in energy. The true situation is more complex and we should consider rays which are non-paraxial and particles which have a small spread in energy. Particle lenses are subject to the same optical aberrations as photon lenses and the effects can be much worse because it is not possible to use materials of different dispersion to reduce chromatic aberration or to grind surfaces of special forms to reduce spherical and other aberrations. 5.1 Spherical Aberration The handling of particle beams is usually an axis-centred problem so Spherical Aberration is normally much more serious than the off-axis aberrations, such as coma, astigmatism and distortion. We shall therefore consider only rays which cross the axis at some point. Figure 5.1 is a schematic diagram showing the emergent asymptote corresponding to a ray incident from an axial object point. For meridional rays, the relationship of the radial positions and slopes of the rays at the first and second focal planes of the lens can be expressed, following Verster [6] as r 2 = − r 1 f 2 + m 13 r 3 1 + m 14 r 2 1 r 1 f 2 + m 15 r 1 r 1 f 2 2 + m 16 r 1 f 2 3 + q 11 r 5 1 + q 12 r 4 1 r 1 f 2 + q 13 r 3 1 r 1 f 2 2 + q 14 r 2 1 r 1 f 2 3 + q 15 r 1 r 1 f 2 4 + q 16 r 1 f 2 5 + · · · ( 5 . 1 a) − r 2 f 1 = r 1 + m 23 r 3 1 + m 24 r 2 1 r 1 f 2 + m 25 r 1 r 1 f 2 2 + m 26 r 1 f 2 3 + q 21 r 5 1 + q 22 r 4 1 r 1 f 2 + q 23 r 3 1 r 1 f 2 2 + q 24 r 2 1 r 1 f 2 3 + q 25 r 1 r 1 f 2 4 + q 26 r 1 f 2 5 + · · · ( 5 . 1 b) 80 Aberrations 81 r i r’ 2 l 2 F 2 q r 1 H 2 H 1 r F 1 o p r’ 1 Figure 5.1 Definition of the ray parameters for the description of the meridional aberration coefficients. remembering that f 1 is negative † . The coefficients, m ij and q ij , are dimensionless and also negative.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 8 Optical Aberration Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after trans-mission through the system does not converge into (or does not diverge from) a single point. Instrument-makers need to correct optical systems to compensate for aberration. Overview Aberrations fall into two classes: monochromatic and chromatic . Monochromatic aberr-ations are caused by the geometry of the lens and occur both when light is reflected and when it is refracted. They appear even when using monochromatic light, hence the name. Chromatic aberrations are caused by dispersion, the variation of a lens's refractive index with wavelength. They do not appear when monochromatic light is used. Monochromatic aberrations • Piston • Tilt • Defocus • Spherical Aberration • Coma • Astigmatism • Field curvature • Image distortion Piston and tilt are not actually true optical aberrations, as they do not represent or model curvature in the wavefront. If an otherwise perfect wavefront is aberrated by piston and tilt, it will still form a perfect, aberration-free image, only shifted to a different position. Defocus is the lowest-order true optical aberration. Chromatic aberrations • Axial, or longitudinal, chromatic aberration ________________________ WORLD TECHNOLOGIES ________________________ • Lateral, or transverse, chromatic aberration Monochromatic aberration The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space. The introduction of simple auxiliary terms, due to C.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Research World(Publisher)
____________________ WORLD TECHNOLOGIES ____________________ Chapter 4 Optical Aberration Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge into (or does not diverge from) a single point. Instrument-makers need to correct optical systems to compensate for aberration. The articles on reflection, refraction and caustics discuss the general features of reflected and refracted rays. Overview Aberrations fall into two classes: monochromatic and chromatic . Monochromatic aberrations are caused by the geometry of the lens and occur both when light is reflected and when it is refracted. They appear even when using monochromatic light, hence the name. Chromatic aberrations are caused by dispersion, the variation of a lens's refractive index with wavelength. They do not appear when monochromatic light is used. Monochromatic aberrations • Piston • Tilt • Defocus • Spherical Aberration • Coma • Astigmatism • Field curvature • Image distortion ____________________ WORLD TECHNOLOGIES ____________________ Piston and tilt are not actually true optical aberrations, as they do not represent or model curvature in the wavefront. If an otherwise perfect wavefront is aberrated by piston and tilt, it will still form a perfect, aberration-free image, only shifted to a different position. Defocus is the lowest-order true optical aberration. Chromatic aberrations • Axial, or longitudinal, chromatic aberration • Lateral, or transverse, chromatic aberration Monochromatic aberration The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space. The introduction of simple auxiliary terms, due to C.
eBook - PDF- Rudolf Kingslake(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
C H A P T E R 5 Spherical Aberration The direct calculation of Spherical Aberration is a simple matter. A meri-dional ray is traced from object to image, passing through the desired zone of a lens, and the image distance L is found. This is compared directly with the /' of a corresponding paraxial ray from the same object point. Then longitudinal Spherical Aberration = LA' = L — I' I. SURFACE CONTRIBUTION FORMULAS The simple relationship just given is often inadequate, both because it gives the aberration as a small difference between two large numbers, and also because it gives no clue as to where the aberration arises. It is therefore much more useful to compute the aberration as the sum of a series of surface contributions. A convenient formula has been given by Delano; 1 the deriva-tion follows from Fig. 53. In this diagram, an entering marginal and paraxial FIG. 53. Spherical Aberration contribution. ray are shown at a spherical surface. The length S is the perpendicular drawn from the paraxial object point P onto the marginal ray. The marginal ray is defined by its Q and U, the paraxial ray by its y and u. Then S = Q — I sin U, hence Su = Qu — y sin U 1 E. Delano, A general contribution formula for tangential rays, J. Opt. Soc. Am., 42, 631 (1952). 101 102 5. Spherical Aberration We now replace u on the right by yc — i and sin U by Qc — sin /, where c is the surface curvature as usual. Multiplying through by n gives Snu = yn sin / — Qni Doing the same thing for the refracted ray and subtracting plain from prime gives S'n'u' — Snu = (Q — Q')ni We write this for every surface and add. After extensive cancellation because (S'n'u') x = (Snu) 2 , we get for k surfaces, (S'nV)* - (Snu), = £ (G -Q')ni (50) Inspection of Fig.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 8 Optical Aberration Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge into (or does not diverge from) a single point. Instrument-makers need to correct optical systems to compensate for aberration. Overview Aberrations fall into two classes: monochromatic and chromatic . Monochromatic aberrations are caused by the geometry of the lens and occur both when light is reflected and when it is refracted. They appear even when using monochromatic light, hence the name. Chromatic aberrations are caused by dispersion, the variation of a lens's refractive index with wavelength. They do not appear when monochromatic light is used. Monochromatic aberrations • Piston • Tilt • Defocus • Spherical Aberration • Coma • Astigmatism • Field curvature • Image distortion ________________________ WORLD TECHNOLOGIES ________________________ Piston and tilt are not actually true optical aberrations, as they do not represent or model curvature in the wavefront. If an otherwise perfect wavefront is aberrated by piston and tilt, it will still form a perfect, aberration-free image, only shifted to a different position. Defocus is the lowest-order true optical aberration. Chromatic aberrations • Axial, or longitudinal, chromatic aberration • Lateral, or transverse, chromatic aberration Monochromatic aberration The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space. The introduction of simple auxiliary terms, due to C.
eBook - PDF- A. B. El-Kareh, J. C. J. El-Kareh(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
X The Spherical Aberration of Electric and Magnetic Lenses In Chapter IX, we discussed the theory of the geometrical aberrations in a detailed manner. The aberration coefficients were evaluated in terms of various parameters of the lens. Those parameters are not always readily available for any specified geometry. The determination of the potential variation along the axis and its derivatives can be computed analytically for only very specialized geometries. However, with the help of a digital computer, the necessary parameters can be evaluated using the relaxation technique. This applies to electric, as well as magnetic, lenses. Most of the available data in the literature were obtained experimentally. We saw in the previous chapter that only the Spherical Aberration does not vanish when the object is on the axis because it is dependent on the aper-ture size only. Most electron optical devices use an electron source which is located on the axis, and therefore the Spherical Aberration is the most im-portant geometric error which appears in this case. It is for this reason that this particular aberration has been studied in detail for various electrostatic and magnetic lenses, and this is why we propose to devote a whole chapter to this subject. 10.1 The Impossibility of Canceling the Spherical Aberration Before determining the coefficient of Spherical Aberration of various lenses, it is probably good to first ask ourselves whether it is possible to design an electrostatic or magnetic lens with rotational symmetry which has a coefficient 50 10.1 Impossibility of Canceling Spherical Aberration 51 of Spherical Aberration equal to zero. With this in mind, let us refer to the ex-pression for the Spherical Aberration derived in the previous chapter.
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