Compound Lenses Thin Lenses in Contact

Compound lenses are lenses which are made from the combination of simple lenses with a common axis.In contrast to compound lens systems, which allow the usage of many lenses along a single common axis, simple lens systems only employ one lens. In this article we are going to study a compound system of thin lens club together with a common axis and derive an equivalent focal length of the compound system made up of two refracting spherical thin lenses. Compound system can be formed by using a series of thin lenses provided having a common axis, but in this article we will discuss the compound system of two lenses and then generalise the formula. We will also discuss applications of compound lenses in real life.

Introduction

When two or more simple lenses joined together to form a system with a common axis is called a Compound lens system. The lenses involved in the combination can be either concave lens or a convex lens. Since, the lens can be considered to be formed by combining two spherical surfaces of different or same refractive media, we must know the refraction formalism in spherical surfaces and refraction formalism by lens formed two spherical surfaces of different suitable radius of curvature. We will discuss the basic formula involved to derive the formula for equivalent focal length of combination of thin lenses and its applications in practical life. Through the following basics we will develop the basics involved that ones need to know white studying compound lenses formed by combination of thin lenses-

• Refraction by Spherical Surfaces- Consider the geometry of formation of image I of an object O on the principal axis of a spherical surface with centre of curvature C, and radius of curvature R. The rays are incident from medium of refractive index , to another of refractive index . The relation between object and image distance in terms of refractive index of the medium and the radius of curvature of the curved spherical surface is given by following formula-

Where, and are the image and object distances from the spherical surface and R is the radius of curvature.

• Refraction by Lens- Consider the geometry of image formation by a double convex lens formed by two spherical surfaces of radii of curvature and . The image formation can be seen in terms of two steps:The first refracting surface forms the image of the object O. The image acts as a virtual object for the second surface that forms the final image by double convex lens at I. In order to design lenses of desired focal length of suitable radii of curvature and , is given by lens formula as follows-

where , relative refractive index of two spherical surfaces,

Through various approximation applying sign convention, thins lens formula can be derived as-

Where, and are the image and object distance from thin lens, is the focal length of a thin lens.

Compound Lenses

Two thin lenses mounted on a common axis, typically closer to one another or frequently cemented together, are known as Compound lenses. It is basically a combination of thin lenses in contact with the common axis.

To study the formalism of refraction by combination of thin lenses, Consider two thin lenses of focal length and placed in contact with each other. Let the object be placed at a point O beyond the focus of the first lens. The first lens produces an image at . Since image is real, it serves as a virtual object for the second lens, producing the final image at I. It must be noted

that the formation of the image by the first lens is estimated only to determine the position of the final image. In fact, the direction of rays emerging from the first lens gets modified in accordance with the angle at which they strike the second lens. Since the lenses are thin, we assume the optical centres of the lenses to be coincident. Let this central point be denoted by P.

For the image formed by the first lens, we get

———————————-(1)

Where, is the position of image .

For the image formed by the second lens , we get

———————————-(2)

Adding Equation (1) and (2), we get

————————–(3)

If the two lens-system is regarded as equivalent to a single lens of focal length f, we have

————————————-(4)

By comparing equations (3) and (4), we get

————————————(5)

The derivation is valid for any number of thin lenses in contact. If several thin lenses of focal length , , ,……… are in contact, the effective focal length of the combinations of thin lenses are given by

———————-(6)

In terms of power, equation (6) can be written as

————————————–(7)

where P is the net power of the combination of lenses. The summation shown in equation(7) is an algebraic sum of individual powers, so some of the terms on the right side may be positive (for convex lenses) and some negative (for concave lenses). Combination of lenses helps to obtain diverging or converging lenses of desired magnification. It also enhances sharpness

of the image. The total magnification m of the combination is a product of magnification (, , ,…) of individual lenses is given by-

————————————–(8)

Such a system of combination of lenses is commonly used in designing lenses for cameras, microscopes, telescopes and other optical instruments of desired focal length, power and magnification.

Application of Compound lens

• Chromatic aberrations, which are present when using single lenses, are rectified in this application by using a compound lens system.Two wavelengths (usually the extremities of red and violet) are brought into focus by combining the lenses in use.
• Compound lenses are used in various optical instruments like telescope, microscope inorder to reduce defects caused using single lens and to get a proper erect and magnified image.

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