spherical aberration

Examples of spherical aberration in the following topics:

• Aberrations

  • A chromatic aberration, also called achromatism or chromatic distortion, is a distortion of colors .
  • This aberration happens when the lens fails to focus all the colors on the same convergence point .
  • A comatic aberration, or coma, occurs when the object is off-center.
  • These aberrations can cause objects to appear pear-shaped.
  • Spherical aberrations are a form of aberration where rays converging from the outer edges of a lens converge to a focus closer to the lens, and rays closer to the axis focus further.

• The Telescope

  • The potential advantages of using mirrors instead of lenses were a reduction in spherical aberrations and the elimination of chromatic aberrations.
  • With the invention of achromatic lenses in 1733, color aberrations were partially corrected, and shorter, more functional refracting telescopes could be constructed.

• Combinations of Lenses

  • The use of multiple elements allows for the correction of more optical aberrations, such as the chromatic aberration caused by the wavelength-dependent index of refraction in glass, than is possible using a single lens.
  • In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations.
  • An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration.
  • The lens elements are mounted next to each other, often cemented together, and shaped so that the chromatic aberration of one is counterbalanced by that of the other.
  • (a) Chromatic aberration is caused by the dependence of a lens's index of refraction on color (wavelength).

• Spherical and Plane Waves

  • Spherical waves come from point source in a spherical pattern; plane waves are infinite parallel planes normal to the phase velocity vector.
  • In 1678, he proposed that every point that a luminous disturbance touches becomes itself a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time.
  • Since the waves all come from one point source, the waves happen in a spherical pattern.
  • All the waves come from a single point source and are spherical .
  • When waves are produced from a point source, they are spherical waves.

• Triple Integrals in Spherical Coordinates

  • When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
  • When the function to be integrated has a spherical symmetry, it is sensible to change the variables into spherical coordinates and then perform integration.
  • It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
  • Points on $z$-axis do not have a precise characterization in spherical coordinates, so $\theta$ can vary from $0$ to $2 \pi$.
  • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.

• Cylindrical and Spherical Coordinates

  • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.
  • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:
  • Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.

• Gravitational Attraction of Spherical Bodies: A Uniform Sphere

  • The Shell Theorem states that a spherically symmetric object affects other objects as if all of its mass were concentrated at its center.
  • For highly symmetric shapes such as spheres or spherical shells, finding this point is simple.
  • A spherically symmetric object affects other objects gravitationally as if all of its mass were concentrated at its center,
  • If the object is a spherically symmetric shell (i.e., a hollow ball) then the net gravitational force on a body inside of it is zero.
  • That is, a mass $m$ within a spherically symmetric shell of mass $M$, will feel no net force (Statement 2 of Shell Theorem).

• Three-Dimensional Coordinate Systems

  • Often, you will need to be able to convert from spherical to Cartesian, or the other way around.
  • The spherical system is used commonly in mathematics and physics and has variables of $r$, $\theta$, and $\varphi$.
  • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.

• Introduction to Spherical and Cylindrical Harmonics

  • In this section we will apply separation of variables to Laplace's equation in spherical and cylindrical coordinates.
  • Spherical coordinates are important when treating problems with spherical or nearly-spherical symmetry.
  • To a first approximation the earth is spherical and so is the hydrogen atom, with lots of other examples in-between.
  • On the other hand, if we tried to use Cartesian coordinates to solve a boundary value problem on a spherical domain, we couldn't represent this as a fixed value of any of the coordinates.
  • Obviously this would be much simpler if we used spherical coordinates, since then we could specify boundary conditions on, for example, the surface $x = r \cos \phi \sin \theta$ constant.

• Spherical Distribution of Charge

  • The charge distribution around a molecule is spherical in nature, and creates a sort of electrostatic "cloud" around the molecule.
  • This distribution around a charged molecule is spherical in nature, and creates a sort of electrostatic "cloud" around the molecule.
  • The attraction or repulsion forces within the spherical distribution of charge is stronger closer to the molecule, and becomes weaker as the distance from the molecule increases.
  • Describe shape of a Coulomb force from a spherical distribution of charge