Astigmatism, Spherical aberration and Comatic Aberration

spherical Aberrations

when the aperture of lens becomes relatively longer than the focal length, the marginal and paraxial rays converges at different point in the principle axis. The marginal rays are focus at the near point of the lens where as the paraxial rays are converged at the far point from the lens, this kind of defect produced by spherical surface is called spherical aberrations.

In the figure above, the marginal rays are focused at the point I, which is very close to the lens. The paraxial rays are focused at another point I’, which is relatively large distance a part from the lens.

Cause of spherical aberrations

The spherical aberration produced by the lens based on the fact that different annular zones of the lens have different focal length.

A system of plano-convex lens is said to be free from spherical aberration, if their different in focal length is equal to the separation between them i.e. \(d = f_1 - f_2\)

Removal of this defect:

1. By using a suitable combination of concave and convex lens.
2. By using stop or aperture( hole) of the suitable diameter. By using the stop or aperture, we can block the paraxial (or marginal ) rays then the image is formed only due to the marginal ( or paraxial rays ) which his free from the spherical aberration. Here, but the image is less intensive.
3. By using two plano-convex lens having equal deviation power

Let us consider two plano-convex lens having deviation power \(\delta_1\) and \(\delta_2\)\. Then net deviation is given by $$\delta=\delta_1+\delta_2$$

Here, we know that spherical aberration is directly proportional to the square of deviation produced by the lens. i.e. spherical aberrations \(\propto \delta^2\)

$$=(\delta_1+\delta_2)^2$$ $$=(\delta_1-\delta_2)^2+4\delta_1 \delta_2$$

Therefore, spherical aberrations will be minimum when \(\delta_1 = \delta_2\)

So, by using two plano- convex lenses having equal deviation power, spherical aberration can be minimized.

4.By using two plano-convex lenses at the distance equal to the difference of their focal length.

Let us consider c be the position of the image produced by the lens \(l_1\) in the absence of the lens \(l_2\). When the lens \(l_2\) is kept in between \(l_1\) and c , then the image c of the lens \(l_1\) acts as the virtual object for the lens \(l_2\) and produces final image at B.

Here, for the lens \(l_2\), DC = object distance = u = -(\(f_1\)-d) and

DB = image distance = v = \(\frac{1}{2}\) DC = \(\frac{1}{2}(f_1-d)\)

And hence \(f_1 = f_2\)

Then we can put these values in the formula, which is free from the spherical aberration.

$$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$$ which implies that; d = \(f_1 – f_2\)

Therefore, by using two plano-convex lenses at the distance equal to the difference of their focal length, image will be free from the spherical aberration.

5.By using a specifying constructed lens called crossed lens, image will be free from the spherical aberration.

Comatic Aberration( Coma ):

In this kind of defect, the point object is situated away from the principle axis. It is similar to the spherical aberrations in the sense that the defect produced for both is due to the unequal magnification produced by the lens for the light waves incident at different points of the lens. The image of the point object are seen as the comet shaped , so that the defect is known as coma. The comatic images are spreaded along the perpendicular direction of the principle axis. To describe the comatic aberration, let us make a light beam incident on lens on the point object not situated in the principle axis.

In the figure, the light rays incident at different points of the lens are focused at different points forming comatic images [the images are shown in the figure (ii)]. This kind of defect can be removed by (i) choosing suitable radii of curvature.

(ii) Using suitable construction of lenses

The system of lens which is used to remove the comatic aberration is called aplantic combination.

Astigmatism:

In this kind of defect, the point object is not situated on the principle axis as in the comatic aberration. This defects also arises due to the unequal magnification produced by the lens for the light waves incident on different plane. The spreaded image are seen along the direction of principle axis.

To illustrate the defect, let us converge the beam of light on a convex lens from a point object not situated on the principle axis as in the diagram below.

The images produced in Astigmatism are shown in the diagram above (figure (ii) ). The plane \(OM_1M_2\) is called tangential plane and the plane \(OS_1S_2\) which is perpendicular to the former plane is called sagittal plane. The light ray sincident on the tangential plane are focussed at the near point of the lens and the light rays incident on the sagittal plane are focussed at the far point from the lens. This kind of defect can beremoved by

(1) using stop or obstacle property.

(2) Choosing suitable radii of curvature.

References:

Adhikari, P.B, Daya Nidhi Chhatkuli and Iswar Prasad Koirala. A Textbook of Physics. Vol. II. Kathmandu: Sukunda Pustak Bhawan, 2012.

Jenkins, F.A and H.E White. Fundamental of optics. New York (USA): McGraw-Hill Book Co, 1976.

wood, R.W. Physical Optics. New York (USA): Dover Publication , 1934.

(3) Using suitable combination of lenses which his called Anastigmatic combination.