APPENDIX C
LENSES
In this small volume it is not desirable, neither is it intended, to give an exhaustive treatment on the subject of lenses and their action, but as optics plays an important part in the transmission of photographs, both by wireless and over ordinary conductors, the following notes relating to a few necessary principles have been included as likely to prove of interest.
Light always travels in straight lines when in a medium of uniform density, such as water, air, glass, etc., but on passing from one medium to another, such as from air to water, or air to glass, the direction of the light rays is changed, or, to use the correct term, refracted. This refraction of the rays of light only takes place when the incident rays are passed obliquely; if the incident rays are perpendicular to the surface separating the two media they are not refracted, but continue their course in a straight line.
All liquid and solid bodies that are sufficiently transparent to allow light rays to pass through them possess the power of bending or refracting the rays, the degree of refraction, as already explained, depending upon the nature of the body.
The law relating to refraction will perhaps be better understood by means of the following diagram. In Fig. 64 let the line AB represent the surface of a vessel of water. The line CD, which is perpendicular to the surface of the
water, is termed the normal, and a ray of light passed in this direction will continue in a straight line to the point E. If, however, the ray is passed in an oblique direction, such as ND, it will be seen that the ray is bent or refracted in the direction DM; if the ray of light is passed in any other oblique direction, such as JD, the refracted ray will be in the direction DK. The angle NDC is called the angle of incidence and MDE the angle of refraction. If we measure accurately the line NC, we shall find that it is 11/3, or more exactly 1.336, times greater than the line EM. If we repeat this measurement with the lines JH and PK we shall find that the line JH also bears the proportion of 1.336 to the line PK. The line NC is called the sine of the angle of incidence NDC, and EM the sine of the angle of refraction MDE.
Therefore in water the sine of the angle of incidence is to the sine of the angle of refraction as 1.336 is to 1, and this is true whatever the position of the incident ray with respect to the surface of the water. From this we say that the sines of the angles of incidence and refraction have a constant proportion or ratio to one another.
The number 1.336 is termed the refractive index, or coefficient, or the refractive power of water. The refractive power varies, however, with other fluids and solids, and a complete table will be found in any good work on optics.
Glass is the substance most commonly used for refracting the rays of light in optical work, the glass being worked up into different forms according to the purpose for which it
is intended. Solids formed in this way are termed lenses. A lens can be defined as a transparent medium which, owing to the curvature of its surfaces, is capable of converging or diverging the rays of light passed through it. According to its curvature it is either spherical, cylindrical, elliptical, or parabolic. The lenses used in optics are always exclusively spherical, the glass used in their construction being either crown glass, which is free from lead, or flint glass, which contains lead and is more refractive than crown glass. The refractive power of crown glass is from 1.534 to 1.525, and of flint glass from 1.625 to 1.590. Spherical surfaces in combination with each other or with plane surfaces give rise to six different forms of lenses, sections of which are given in Fig. 65.
All lenses can be divided into two classes, convex or converging, or concave or diverging. In the figure, b, c, g are converging lenses, being thicker at the middle than at the borders, and d, e, f, which are thinner at the middle, being diverging lenses. The lenses e and g are also termed meniscus lenses, and a represents a prism. The line XY is the axis or normal of these lenses to which their plane surfaces are perpendicular.
Let us first of all notice the action of a ray of light when passed through a prism. The prism, Fig. 66, is represented by the triangle BBB, and the incident ray by the line TA.
Where it enters the prism at A its direction is changed and it is bent or refracted towards the base of the prism, or towards the normal, this being always the case when light passes from a rare medium to a dense one, and where the light leaves the opposite face of the prism at D it is again refracted, but away from the normal in an opposite direction to the incident ray, since it is passing from a dense to a rare medium. The line DP is called the emergent or refracted ray. If the eye is placed at T, and a bright object at P, the object is seen not at P, but at the point H, since the eye cannot follow the course taken by the refracted rays. In other words, objects viewed through a prism always appear deflected towards its summit.
In considering the action of a lens we can regard any lens as being built up of a number of prisms with curved faces in contact. Such a lens is shown in Fig. 67, the light rays being refracted towards the base of the prisms or towards the normal, as already explained; while the top half of the lens will refract all the light downwards, the bottom half will act as a series of inverted prisms and refract all the light upwards.
If a beam of parallel light—such as light from the sun—be passed through a double convex lens L, Fig. 68, we shall find that the rays have been refracted from their parallel course and brought together at a point F. This point F is
termed the principal focus of the lens, and its distance from the lens is known as the focal length of that lens. In a double and equally convex lens of glass the focal length is equal to the radius of the spherical surfaces of the lens. If the lens is a plano-convex the focal length is twice the radius of its spherical surfaces. If the lens is unequally convex the focal length is found by the following rule: multiply the two radii of its surfaces and divide twice that product by the sum of the two radii, and the quotient will
be the focal length required. Conversely, by placing a source of light at the point F the rays will be projected in a parallel beam the same diameter as the lens. If, however, instead of being parallel, the rays proceed from a point farther from the lens than the principal focus, as at A, Fig. 69, they are termed divergent rays, but they also will be brought to a focus at the other side of the lens at the point a. If the source of light A is moved nearer to the principal focus of the lens to a point A1 the rays will come to a focus at the point a1, and similarly when the light is at A2 the rays will come to a focus at the point a2. It can be found by direct experiment that the distance fa increases in the same proportion as AF diminishes, and diminishes in the same proportion as AF increases. The relationship which exists between pairs of points in this manner is termed the conjugate foci of a lens, and though every lens has only one principal focus, yet its conjugate foci are innumerable.
The formation of an image of some distant object in its principal focus is one of the most useful properties of a convex lens, and it is this property that forms the basis of several well-known optical instruments, including the camera, telescope, microscope, etc.
If we take an oblong wooden box, AA, and substitute a sheet of ground glass, C, for one end, and drill a small pinhole, H, in the centre of the other end opposite the
glass plate, we shall find that a tolerably good image of any object placed in front of the box will be formed upon the glass plate. The light rays from all points of the object, BD, Fig. 70, will pass straight through the hole H, and illuminate the ground glass screen at points immediately opposite them, forming a faint inverted image of the object BD. The purpose of the hole H is to prevent the rays from any one point of the object from falling upon any other point on the glass screen than the point immediately opposite to it, therefore the smaller we make H, the more distinct will be the image obtained. Reducing the size of H in order to produce a more distinct image has the effect of causing the image to become very faint, as the smaller the hole in H, the smaller the number of rays that can pass through from any point of the object. By enlarging the hole H gradually, the image will become more and more indistinct until such a size is reached that it disappears altogether.
If in this enlarged hole we place a double convex lens, LL, Fig. 71, whose focal length suits the length of the box, the image produced will be brighter and more distinct than that formed by the aperture, H, since the rays which proceed from any point of the object will be brought by the lens to a focus on the glass screen, forming a bright
distinct image of the point from which they come. The image owes its increased distinctness to the fact that the rays from any one point of the object cannot interfere with the rays from any other point, and its increased brightness to the great number of rays that are collected by the lens from each point of the object and focussed in the corresponding point of the image. It will be evident from a study of Fig. 71 that the image formed by a convex lens must necessarily be inverted, since it is impossible for the rays from the end, M, of the object to be carried by refraction to the upper end of the image at n. The relative positions of the object and image when placed at different distances from the lens are exactly the same as the conjugate foci of light rays as shown in Fig. 69.
The length of the image formed by a convex lens is to the length of the object as the distance of the image is to the distance of the object from the lens. For example, if a lens having a focal length of 12 inches is placed at a distance of 1000 feet from some object, then the size of the image will be to that of the object as 12 inches to 1000 feet, or 1000 times smaller than the object; and if the length of the object is 500 inches, then the length of the image will be the 1/1000th part of 500 inches, or 1/2 inch.
The image formed by the convex lens in Fig. 71 is known as a real image, but in addition convex lenses possess the property of forming what are termed virtual images. The distinction can be expressed by saying, real images are those formed by the refracted rays themselves, and virtual images those formed by their prolongations. While a real image formed by a convex lens is always inverted and smaller than the object, the virtual image is always erect and larger than the object. The power possessed by convex lenses of forming virtual images is made use of in that useful but common piece of apparatus known as a reading or magnifying glass, by which objects placed within its focus are made larger or magnified when viewed through it; but in order to properly understand how objects seem to be brought nearer and apparently increased in size, we must first of all understand what is meant by the expression, the apparent magnitude of objects.
The apparent magnitude of an object depends upon the angle which it subtends to the eye of the observer. The image at A, Fig. 72, presents a smaller angle to the eye than the angle presented by the object when moved to B, and the image therefore appears smaller. When the object is moved to either B or C, it is viewed under a much
greater angle, causing the image to appear much larger. If we take a watch or other small circular object and place it at A, which we will suppose is a distance of 50 yards, we shall find that it will be only visible as a circular object, and its apparent magnitude or the angle under which it is viewed is then stated to be very small. If the object is now moved to the point B, which is only 5 feet from the eye, its apparent magnitude will be found to have increased to such an extent that we can distinguish not only its shape, but also some of the marking. When moved to within a few inches from the eye as at C, we see it under an angle so great that all the detail can be distinctly seen. By having brought the object nearer the eye, thus rendering all its parts clearly visible, we have actually magnified it, or made it appear larger, although its actual size remains exactly the same. When the distance between the object and the observer is known, the apparent magnitude of the object varies inversely as the distance from the observer.
Let us suppose that we wish to produce an image of a tree situated at a distance of 5000 feet. At this distance the light rays from the tree will be nearly parallel, so that if a lens having a focal length of 5 feet is fastened in any convenient manner in the wall of a darkened room the image will be formed 5 feet behind the lens at its principal focus. If a screen of white cardboard be placed at this point we shall find that a small but inverted image of the tree will be focussed upon it. As the distance of the object is 5000 feet, and as the size of the received image is in proportion to this distance divided by the focal length of the lens, the image will be as 5000 ÷ 5, or 1000 times smaller than the object.
If now the eye is placed six inches behind the screen and the screen removed, so that we can view the small image distinctly in the air, we shall see it with an apparent magnitude as much greater than if the same small image were equally far off with the tree, as 6 inches is to 5000
feet, that is 10,000 times. Thus we see that although the image produced on the screen is 1000 times less than the tree from one cause, yet on account of it being brought near to the eye it is 10,000 times greater in apparent magnitude; therefore its apparent magnitude is increased as 10,000 ÷ 1000, or 10 times. This means that by means of the lens it has actually been magnified 10 times. This magnifying power of a lens is always equal to the focal length divided by the distance at which we see small objects most distinctly, viz. 6 inches, and in the present instance is 60 ÷ 6, or 10 times.
When the image is received upon a screen the apparatus is called a camera obscura, but when the eye is used and sees the inverted image in the air, then the apparatus is termed a telescope.
The image formed by a convex lens can be regarded as a new object, and if a second lens is placed behind it a second image will be formed in the same manner as if the first image were a real object. A succession of images can thus be formed by convex lenses, the last image being always treated as a fresh object, and being always an inverted image of the one before. From this it will be evident that additional magnifying power can be given to our telescope with one lens by bringing the image nearer the eye, and this is accomplished by placing a short focus lens between the image and the eye. By using a lens having a focal length of 1 inch, and such a lens will magnify 6 times, the total magnifying power of the two lenses will be 10 × 6 = 60 times, or 10 times by the first lens and 6 times by the second. Such an instrument is known as a compound or astronomical telescope, and the first lens is called the object glass and the second lens the magnifying glass, or eye-piece.
We are now in a position to understand how virtual images are formed, and the formation of a virtual image by means of a convex lens will be readily followed from a
study of Fig. 73. Let L represent a double convex lens, with an object, AB, placed between it and the point F, which is the principal focus of the lens. The rays from the object AB are refracted on passing through the lens, and again refracted on leaving the lens, so that an image of the object is formed at the eye, N. As it is impossible for the eye to follow the bent rays from the object, a virtual image is formed and is seen at A1B1, and is really a continuation of the emergent rays. The magnifying power of such a lens may be found by dividing 6 inches by the focal length of the lens, 6 inches being the distance at which we see small objects most distinctly. A lens having a focal length of 1/4 inch would magnify 24 times, and one with a focal length of 1/100th of an inch 600 times, and so on. The magnifying power is greater as the lens is more convex and the object near to the principal focus. When a single lens is applied in this manner it is termed a single microscope, but when more than one lens is employed in order to increase the magnifying power, as in the telescope, then the apparatus is termed a compound microscope.
Unlike a convex lens, which can form both real and virtual images, a concave lens can only produce a virtual image; and while the convex lens forms an image larger
than the object, the concave lens forms an image smaller than the object. Let L, Fig. 74, represent a double concave lens, and AB the object. The rays from AB on passing through the lens are refracted, and they diverge in the direction RRRR, as if they proceeded from the point F, which is the principal focus of the lens, and the prolongations of these divergent rays produce a virtual image, erect and smaller than the object, at A1B1. The principal focal distance of concave lenses is found by exactly the same rule as that given for convex lenses.
Up to the present we have assumed that all the rays of light passed through a convex lens were brought to a focus at a point common to all the rays, but this is really only the case with a lens whose aperture does not exceed 12°. By aperture is meant the angle obtained by joining the edges of a lens with the principal focus. With lenses having a larger aperture the amount of refraction is greater at the edges than at the centre, and consequently the rays that pass through the edges of the lens are brought to a focus nearer the lens than the rays that pass through the centre. Since this defect arises from the spherical form of the lens it is termed spherical aberration, and in lenses that
are used for photographic purposes the aberration has to be very carefully corrected.
The distortion of an image formed by a convex lens is shown by the diagram, Fig. 75. If we receive the image upon a sheet of white cardboard placed at A, we shall find that while the outside edges will be clear and distinct, the inside will be blurred, the reverse being the case when the cardboard is moved to the point B.
Aberration is to a great extent minimised by giving to the lens a meniscus instead of a biconvex form, but as it is desirable to reduce the aberration to below once the
thickness of the lens, and as this cannot be done by a single lens, we must have recourse to two lenses put together. The thickness of a lens is the difference between its thickness at the middle and at the circumference. In a double convex lens with equal convexities the aberration is 167/100ths of its thickness. In a plano-convex lens with the plane side turned towards parallel rays the aberration is 41/2 times its thickness, but with the convex side turned towards parallel rays the aberration is only 117/100ths of its thickness.
By making use of two plano-convex lenses placed together as at Fig. 76, the aberration will be one-fourth of that of a single lens, but the focal length of the lens, L1, must be half as much again as that of L. If their focal lengths are equal the aberration will only be a little more than half reduced. Spherical aberration, however, may be entirely destroyed by combining a meniscus and double convex lens, as shown in Fig. 77, the convex side being turned to the eye when used as a lens, and to parallel rays when used as a burning glass or condenser.