Optical Instruments

Power of a lens Optometrists, instead of using focal length, use the reciprocal of the focal length to specify the strength of eyeglasses or contact lenses. This is called power. P = f -1

Diopter (D) The unit for lens power is the diopter (D), which is an inverse meter (m -1 ) For example, a 20. cm focal length lens has a power of 1/(0.20 m) or 5.0 m -1. Using the unit diopter, the power is written as 5.0 D.

Linear Magnification Linear magnification is the ratio of the size of an image to the size of an object. It will be represented by a lower case m. m = h i /h o

The Eye

Near Point The closest distance at which the eye can focus clearly is called the near point of the eye. For young adults, the near point is typically 25 cm. As people grow older, the ability of the eye to accommodate is reduced and the near point increases. * The focusing adjustment of the eye is called accommodation.

Far Point A given person’s far point is the farthest distance at which an object can be clearly seen. For the normal eye, the far point may be assumed to be at infinity.

Simple Magnifier One of the most basic optical devices people use is a simple magnifier or magnifying glass.

How large an object appears depends on the size of the image it makes on the retina. The size of the image made on the retina depends on the angle subtended by the object at the eye. When we want to examine an object in greater detail, we bring it closer to our eye so that it subtends a greater angle. However, our eyes can only accommodate up to the near point of 25 cm.

A magnifying glass allows us to effectively place the object closer to our eye so that it subtends a greater angle. * In order to produce a virtual image, the object must be placed between f and the lens.

Angular Magnification The angular magnification (M) of a lens is defined as the ratio of the angle subtended by the object when using the lens to the angle subtended by the object using the unaided eye (with the object located at the near point N) M = Θ′/Θ

Magnification of Simple Magnifier We are interested in two extremes. When the image is located at infinity and when the image is located at the eye’s near point.

Image located at infinity If the eye is relaxed, the object will be at the focal point and the image at infinity. In this case, Θ′ = h/d o = h/f and Θ = h/N Prove M = N/f

Image located at the near point The magnification of a given lens can be increased slightly by moving the lens and adjusting your eye so that it focuses on the image at the eye’s near point. In this case, d i = -N, Θ′ = h/d o and Θ = h/N. Prove M = 1 + N/f

Practice Problem An 8. cm focal length lens is used by a jeweler as a magnifying glass. Estimate the angular magnification when (a) the eye is relaxed (answer = 3) and (b) when the eye is focused at its near point (answer = 4)

The Compound Microscope

Objective Lens and Eyepiece Both microscopes and telescopes make use of an objective lens and an eyepiece. The objective lens is the lens closer to the object. It has a focal length of f o and causes a real image to be formed. The eyepiece (focal length f e ) uses the image formed by the objective lens as its object. The eyepiece gives a virtual image that is larger than the image formed by the objective lens.

Ray Diagram for a Compound Microscope

Telescopes Several types of astronomical telescopes exist. The common refracting type, sometimes called Keplerian, contains two converging lenses located at opposite ends of a long tube.

Ray Diagram for an Astronomical Telescope

Angular Magnification for astronomical telescopes M = -f 0 /f e

Lens Aberrations When we learned about lens ray diagrams, we made a couple of approximations that, in reality, are not true. We assumed that all rays of light would be close to the principal axis, and we assumed that light would pass though the center of the lens without bending. In reality, due to the true nature of lenses, aberrations may occur that result in an imperfect image.

Spherical Aberration Spherical aberration occurs because spherical surfaces are not the ideal shape with which to make a lens, but they are by far the simplest shape to which glass can be ground and polished and so are often used. Spherical aberration causes beams parallel to, but distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image.

Spherical Aberration Diagram

Reducing Spherical Aberration Spherical aberration can be corrected by using non-spherical lens surfaces. However, to grind such a lens is expensive. Spherical aberration is generally minimized by using a series of lenses in combination and only using the central part of the lens.

Chromatic Aberration Spherical aberration will occur for monochromatic light is called a monochromatic aberration. Normal light, however is not monochromatic, but contains all the colours of the spectrum. Chromatic aberration occurs due to the different colours of light passing through a lens simultaneously.

Chromatic aberration occurs because of dispersion – the variations of index of refraction with colour resulting in the different colours of light bending different amounts. For example, blue light bends more than red light as it passes through a prism. So, if white light passes through a lens, the different colours will focus at different points. This is chromatic aberration.

Diagram showing chromatic aberration

Eliminating Chromatic Aberration Chromatic aberration can be eliminated for any two colours (and greatly reduced for all others) by the use of two lenses made of different materials with different indices of refraction. Normally, one lens is converging and one is diverging and they are cemented together. This combination is called an achromatic doublet.

Diagram for achromatic doublet