 # Aberration

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Yerassyl Balkybek

Aberrations

Abstract

Aberration is a failure of a mirror or lens to produce a true image according to the derived simple formulas. However, their such kind of imperfections can be minimized by using another one lens’s aberration as a tool to cancel out the others aberrations. There are many types of aberrations, and in this experiment, we examined six main types of them: spherical aberrations, curvature of field, astigmatism, coma, distortion and chromatic aberration. In this experiment we have experimented all of this six major aberrations and presented our results.

Introduction

Aberration can be considered as firstly, monochromatic aberrations, in which mainly spherical aberrations, curvature of field, astigmatism, coma, distortion, and secondly chromatic. The reason for that is the five monochromatic aberrations can be obtained using only monochromatic beams, but chromatic aberration needs many wavelengths. In this lab report, we start discussing the theory of each type of aberrations and move on to the experimental procedure, data and its results and discussion.

Theory

Spherical aberration

In order to make simple formulas for the image distance, one can use paraxial approximations, which is a first order approximation in Taylor expansion of like such function as sine, tangent.  For example, if in fig. 1(a) is a true ideal image formation, then in fig.1 (b) there is an actual image formation what we may actually encounter in our labs. As the fig. 1 (b) shows, parallel beams are failed to focus at one point, this is sometimes called longitudinal spherical aberration. In fig. (2), a point source O has different image locations depending on the rays how far they are passing through the lens’s from the center.

Fig.1 (a) Ideal converging lens                                    Fig. 1 (b) Actual lens    

Fig. (2) Two rays passing through different regions of the lens 

Curvature of field

Curvature of field occurs when a planar object is formed as a curved image. As you can see in fig. 3, an object BA is formed like a curved AB. As you can see, as the angle AOC becomes more and more large, the curvature at the edges increases. Curvature varies as the square of the image height. The more illustrative image can be seen in fig. 4 where an image of a rectangular “+” is presented.

Fig. 3 Curvature of field caused by a lens. 

Fig.4. Curvature of “+” shaped object. 

Actually, a positive lenses form an inward curving fields, and negative lenses form outward curving fields. In our lab, we will use positive lens and have an image of inward curving fields.

Astigmatism

Astigmatism occurs when an object which is not in the optical axis, fail to produce unique image, or fail to focus at a single point.  As you can see from fig. 5, there are two perpendicular planes, one is vertical – meridional/tangentional, and the second is – sagittal, which is along the chief and central horizontal line of the lens. As you can see in the fig. 6, behind the lens an elliptical bundle of rays are formed. The size of them will decrease until rays in meridional plane will be focused at one point and a horizontal line will be produced because of the sagittal plane rays. This is called primary image, and after this point, a small circle is created, called circle of least confusion. Finally, this circle turns to elliptical with vertical major axis, and it turns to vertical line, called secondary image, when rays in meridional plane converge to a point. It is assumed that in the case of astigmatism, the location of the best focus is at the location of the circle of least confusion.

Fig.5. The two sagittal and meridional planes. 

Fig. 6. Image formation in case of astigmatism. 

Coma

Coma is another type of aberrations, in which when an off axis object is imaged by the lens, rays passing through the outer regions of the lens fails to focus at one point. If we consider the rays that are passing through the lens at fixed distance from the center of the lens, then they will be imaged as a circle. The the size of each circle will be increased as rays pass outer margins of the lens, and shifted towards the center if a lens is negative, and away from the center if a lens is positive, and forming in total a “V” shaped image like a comet as in the fig. 6 (b).

Fig. 6. (a) Coma                                                       Fig. 6. (b) Imagination of Coma

Distortion

Distortion arises when a lens is not thin enough and the magnification of an object will change as the distance between an object point and optical axis will change, i.e. different areas of a lens have also different focal length. For example, in many cases, the distance of any point of true image from it’s optical axis can be imagined as it is factored by k, i.e. if initially the element point of true image was at distance r from the optical axis, then because of the distortion it became k r. If k>1 then the image suffers from pincushion, but, if 0 < k < 1, barrel distortion appears. This can be visualized in fig. 7 below, in case of pincushion, as you can see, rectangle is slightly enlarged, and outer lines become more curved inward, but, in case of barrel distortion, rectangles become slightly diminished, outer lines becoming more and more curving outward as its distance from the center increases.

Fig 7. Distortion effects on a rectangular image

Chromatic Aberration

Chromatic aberration occurs because of the dispersion, i.e. because of the different refractive indexes at different wavelengths. Since blue light has higher refractive index than the red light, it focuses nearer to the lens than the red light. Therefore, when a white light passes through the lens, it can cause a little blurred image.

Fig.8. Chromatic aberration

Experimental procedure, data analysis and discussion.

Spherical aberrations

In this part of the experiment, we used He-Neon laser, plano-convex lens (f = 7.5 cm), a small millimeter ruler as an object, beam expander (f = 4.5mm), white screen and setup them as shown in schematic fig.8.

Fig. 9. Experimental setup 

Moreover, we positioned the lens at a position in optical axis, x = 50 cm, and distance between the lens and the object was 10cm, as it was required, and recorded the positions of the clear image, with or without the iris behind the lens. Since the clear image was formed in some distance intervals, we have included them in table 1 as the minimum position, maximum position and most probable position of the clearer image, and average of them. Finally, we calculated the focal length for that lens.

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