Aberration And Chromatic aberration

Introduction

The general equations of optics like lens formula and mirror formula are obtained by assuming the angle subtended by the incident and emergent rays to be small. Derivation of the image formed by the lens from its actual shape, size and position is called aberration. Aberration produced by the lens is due to the refraction of light but not the defective construction of the lens. There are two types of aberrations. They are:

1. Monochromatic aberration
2. Chromatic aberration

Monochromatic aberration:

Aberration produced by the lens when a monochromatic light (i.e. light of single wavelenght or single color light) is used is called monochromatic aberration.

There are different types of aberration, such as

1. i) Spherical aberration
2. ii) comatic aberration (coma)

iii) curvature of field

1. iv) Astigmatic aberration (Astigmatism)
2. v) Distortion

Refraction at Spherical Surface

When a aperture of a lens is relatively large compared to the focal length, the different annular zones have different focal length. So, if a point object is kept at some point on the principle axis of the lens, the marginal rays come to focus at a point \(I_m\) closer to the lens than the focus \(I_p\) of paraxial rays. This type of aberration is called spherical aberration.

It is clear that from the above figure, the image due to marginal rays (1, 1) lie at shorter distance than due to paraxial rays (2, 2). This means the image is not a single sharp point focus. If the screen is kept at \(I_m\), the outer portion of the point is in focus but if the screen is placed at \(I_p\), inner portion is in focus. If the screen is kept at AB, we get a circular patch of image. This circular patch is called circle of least confusion and corresponds to the best image. Here, \(I_m I_p\) is the measure of longitudinal spherical aberrations while the radius of the circle of least confusion gives the measure of lateral spherical aberrations.

Here, the above figure (2.1) shows the spherical aberrations in convex lens whereas the figure (2.2) shows the spherical aberration in concave lens. Also, the spherical aberrations of convex lens is said to be positive and that of concave is negative.

Chromatic Aberration

When a beam of light is passed through a thin lens, it splits into constituent colors. In this case, thin lens acts as the collection of many prisms. The seven different rays are focused at seven different points. If the light ray is incident from the point object, then seven different colorful images are produced. The inability of the lenses which can’t converge the dispersed beams at a single point is called chromatic aberration of the lenses. It arises due to the dispersion of the light through the lens.

Expression for Longitudinal Chromatic Aberrations for an Object at infinity

Consider a point object O from which a light ray is incident on a point having focal length f. The light ray disperses into 7 different colors so that the image of the point object ranges from \(f_v\) to \(f_r\) ( where \(f_v\) to \(f_r\) are the focal length for the violet and red colors). The distance between \(f_v\) to \(f_r\) gives the expression for longitudinal chromatic aberrations.

i.e. longitudinal chromatic aberrations = \(f_r\) - \(f_v\)

we know, Lens Makers formula as $$\frac{1}{f}=(\mu-1)\biggl(\frac{1}{R_1}+\frac{1}{R_2}\biggr)$$

$$\biggl(\frac{1}{R_1}+\frac{1}{R_2}=\frac{1}{f(\mu-1)}\dotsm(1)$$

For red color, $$\frac{1}{f_r}=(\mu_r-1)\biggl(\frac{1}{R_1}+\frac{1}{R_2}\biggr)$$

$$\therefore\;\; \frac{1}{f_r}=\frac{(\mu_r-1)}{f(\mu-1)}\dotsm(2)$$ [\therefore\) \;using (1)]

Similarly for violet color, $$\frac{1}{f_v}=\frac{(\mu_v-1)}{f(\mu-1)}\dotsm(3)$$

Subtracting equation (2) from (3)

$$\frac{1}{f_v}-\frac{1}{f_r}=\frac{\mu_v-1}{f(\mu-1)}-\frac{\mu_r-1}{f(\mu-1)}$$

$$\therefore\;\;\frac{f_r-f_v}{f_r f_v}=\frac{\mu_v-\mu_r}{f(\mu-1)}\dotsm(4)$$

Let f be the geometric mean of the \(f_v\) and \(f_r\) then, f = \(\sqrt{f_r f_v}\)

Which implies that \(f^2 = f_r f_v\)

Then equation (4) becomes $$\frac{f_r-f_v}{f^2}=\frac{\mu_v-\mu_r}{f(\mu-1)}$$

Or, \(f_r – f_v = \biggl[\frac{\mu_v-\mu_r}{(\mu – 1)}\biggr]f\)

So therefore, \(f_r – f_v = \omega f\), where \(\omega\) is the dispersive power of the lens. Where, \(\omega = \frac{\mu_v-\mu_r}{\mu-1}\)

Therefore, longitudinal chromatic aberrations is just the product of dispersive power and focal length of the lens.

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.