Astigmatism is the result of the failure of a single zone of the lens to focus the image of an off-axis point at a single point. In the figure, we see two planes perpendicular to each other passing through the optical axis. These planes are known as the meridian plane and the sagittal plane, the meridian plane being the one containing the off-axis object point. Skew rays, rays not in the meridian plane are focused farther away from the lens than those lying in the plane. In either case, the rays do not meet in a point focus but as lines perpendicular to each other. Intermediate between these two positions, the images are elliptical in shape.

In an ideal lens, light from any given point of an object would pass through the lens and come together at a single point in the image plane. Contrary to ideal lenses, real lenses do not focus light to a single point. These deviations from the idealized lens performance are known as an aberration of the lens.

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Circular polarization

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by R Paschotta · Cited by 2 — In the simplest case, a light beam is linearly polarized, which means that the electric field oscillates in a certain linear direction perpendicular to the beam ...

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Elliptically polarizedformula

When the light of a single wavelength is considered, there are five monochromatic aberrations to be considered and they are:

The name “Coma” is derived from the fact that a point image is blurred into a comet shape when rays from an off-axis object point are imaged by different zones of the lens. In spherical aberration, the images of an object point that fall on a plane perpendicular to the optical axis are circular in the shape of varying sizes and superimposed about a common centre. In a coma, the images of an off-axis object point are circular, of varying size, but displaced with respect to each other.

Elliptical polarization example

When the light is not monochromatic (not of a single wavelength), a sixth aberration is found in lenses but not in mirrors and is named chromatic aberration.

When the sun is at a low angle in the sky, the sunlight reflecting off the surface of water is nearly 100% horizontally polarized because the angle of incidence is close to the Brewster angle.  Glare-reducing sunglasses are coated with a polarizer with a vertical transmission axis and therefore block the reflected light.

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The image shows two images, one resulting from a central cone of rays and the other from a cone passing through the rim.

Elliptically polarizedexamples

A quarter-wave plate δ = π/2 can be used to convert linearly polarized light to circularly polarized light.  The incident linearly polarized light must be oriented at 45o to the wave plate's axes.  A half-wave plate δ = π can be used to rotate the plane of linearly polarized light.  The angle of rotation is 2θ, where θ is the angle between the angle of polarization and the wave plate's fast axis.

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The extraordinary ray violates both Snell’s Law and the Law of Reflection.  It is not necessarily confined to the plane of incidence.  Its speed changes with direction.  The index of refraction for the extraordinary ray is a continuous function of direction.  The index of refraction for the ordinary ray is independent of direction.  When the ordinary index of refraction is plotted against wavelength, the dispersion curve for the ordinary ray is a single unique curve.  The dispersion curve for the extraordinary ray is a family of curves with different curves for different directions.  A ray normally incident on a birefringent crystalline surface will be divided into two rays at the boundary, unless it is in a special polarization state or unless the crystalline surface is perpendicular to an optic axis.  The extraordinary ray will deviate from the incident direction while the ordinary ray will not.  The ordinary ray index n0 and the most extreme extraordinary ray index ne are together known as the principal indices of refraction of the material.  The direction of the lesser index is called the fast axis because light polarized in that direction has the higher speed.

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The electric field vector E can always be resolved into two perpendicular components.  The light is elliptically polarized, then the two components have a constant phase difference, and the tip of the electric field vector traces out an ellipse in the plane perpendicular to the direction of propagation.

In optics, aberration is a property of optical systems such as lenses that results in light being spread out over some region of space rather than being focused to a point. An image-forming optical system with aberration will produce an image that is not sharp. Aberration can be caused due to a variety of reasons such as lens size, material, thickness and position of the object. In this article, let us learn more about aberration and its type.

Elliptically polarizedlight

In spherical aberration, rays of light from a point on the optical axis of a spherical lens do not all meet at the same image point. Rays passing closer to the center are focussed farther away than the rays passing through a circular zone near its rim. A circular cross-section is formed whenever a plane held perpendicular to the optical axis is made to intersect a cone. The area of the cross-section varies with the distance along the optical axis. The smallest size is known as the circle of least confusion. The most spherical aberration-free image is found at this distance.

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Elliptical polarization equation derivation

Elliptically polarizedlight equation

The two beams within the birefringent crystal are referred to as the ordinary and extraordinary ray, respectively.  The polarization of the extraordinary ray lies in the plane containing the direction of propagation and the optic axis, and the polarization of the ordinary ray is perpendicular to this plane.

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Circularlypolarizedlight

The curvature of field and distortion refer to the location of image points with respect to one another. The former three aberrations mentioned so far can be corrected by making corrections in the design of the lens, but these two aberrations could remain. In curvature of the field, the image of a plane object perpendicular to the optical axis will lie on a paraboloidal surface known as the Petzval Surface. Distortion, on the other hand, refers to the deformation of an image. There are two kinds of deformation, namely barrel distortion and pincushion distortion.

The failure of a lens to focus all colours in the same plane is known as chromatic aberration. The refractive index for red is least at the red end of the spectrum, hence the focal length of a lens in the air will be greater for red and green than it would be for blue and violet. Chromatic aberration affects magnification along the optical axis and the axis perpendicular to it. The former is known as longitudinal chromatic aberration and the latter is known as lateral chromatic aberration.

If a beam of linearly polarized monochromatic light enters a birefringent crystal along a direction not parallel to the optical axis of the crystal, the beam will may be divided into two separate beams.  Each will be polarized at right angles to the other, and they will travel in different directions.  The intensity of the original beam will be divided between the two new beams in a manner which depends on the original orientation of the electric field vector with respect to the crystal.  The ratio or the intensities of the two orthogonally polarized beams can have any value.

The image depicts a case of spherical aberration. The most spherical aberration-free image is found at the circle of least confusion.

Linearly polarized light is a special case of elliptically polarized light.  If the light is linearly polarized, then the two components oscillate in phase,  for example Ex = E0xexp(i(kz - ωt)), Ey = E0yexp(i(kz - ωt)), φ = 0.  The direction of E and the direction of propagation define a plane.  The electric vector traces out a straight line.  For example, E = Ei = E0xexp(i(kz - ωt))i.

In other devices the changes in direction of propagation between the two rays is used to separate the incoming beam into two orthogonally polarized beams as in the Wollaston and Thompson beam-splitting prisms.

The figure below shows the trace of the field vector Ex = E0exp(i(kz - ωt)), Ey = E0exp(i(kz - ωt + φ)) in a plane perpendicular to the z-axis when looking towards the source.  (E0x = E0y = E0)

A half-wave plate δ = π can be used to rotate the plane of linearly polarized light.  The angle of rotation is 2θ, where θ is the angle between the angle of polarization and the wave plate's fast axis.

In the Glan-Taylor polarizing prism shown on the right the rejected (ordinary) ray is absorbed by black mounting material in the prism housing.