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The refraction of the eye - Optics

Light is propagated from a luminous point in every plane and in every direction in straight lines; these lines of direction are called rays. Rays travel with the same rapidity so long as they remain in the same medium.

The denser the medium the less rapidly does the ray of light pass through it. Rays of light diverge, and the amount of divergence is proportionate to the distance of the point from which they come; the nearer the source of the rays the more they diverge.

When rays proceed from a distant point such as the sun, it is impossible to show that they are not parallel, and in dealing with rays which enter the eye, it will be sufficiently accurate to assume them to be parallel when they proceed from a point at a greater distance than 6 metres.

A ray of light meeting with a body may be absorbed, reflected, or if it is able to pass through this body it may be refracted.

Reflection

Reflection by a Plane Surface

Reflection takes place from any polished surface, and according to two laws.
1. The angle of reflection- is equal to the angle of incidence.
2nd. The reflected and incident rays are both in the same plane, which is perpendicular to the reflecting surface.

Fig 1


Thus, if A B be the ray incident at B, on the mirror C D, and B E be the ray reflected, the perpendicular P B will divide the angle A B E into two equal parts.
The angle A B P is equal to the angle P B E ; while A B, P B, and E B lie in the same plane.

When reflection takes place from a plane surface, the image is projected backwards to a distance behind the mirror, equal to the distance of the object in front of it, the image being of the same size as the object.

Thus in Fig. 2 the image of the candle c is formed behind the mirror M, at C', a distance behind the mirror equal to the distance of the candle in front of it, an observer's eye placed at E would receive the rays from C as if they came from C'.

Fig 2


The image of the candle so formed by a plane mirror is called a virtual image.

Reflection by a Concave Surface

A concave surface may be looked upon as made up of a number of planes inclined to each other. Parallel rays falling on a concave wiaror are reflected as convergent rays, which meet on the axis at
a point (F, Fig. 3) called the principal focus, midway between the rnirror and its optical centre C. The distance of the principal focus from the mirror is called the focal length of the mirror.

If the luminous point be situated at F, then the diverging rays would be reflected as parallel to each other and to the axis.
If the luminous point is at the centre of the concavity of the mirror (C), the rays return along the same lines, so that the point is its own image.

Fig 3


If the luminous point be at A the focus will be at a, and it is obvious that if the luminous point be moved to a, its focus will be at A; these two points therefore, A and a, bear a reciprocal relation to eachother, and are called conjugate foci.
If the lumiinous point is beyond the centre, its conjugate focus is between the principal focus and the centre.

If the luminous point is between the principal focus and the centre, then its conjugate is beyond the centre ; so that the nearer the luminous point approaches the principal focus, the greater is the distance at which the reflected rays meet.

If tile luminous point be nearer thc mirror than F, the principal focus, the rays will be reflected as divergent, and will therefore never meet: if, however, we continue these diverging rays backwards, they will unite at a point (x) behind the mirror; this point is called the virtual focus, and an observer situated in the path of reflected rays will receive them as if they came from this point.


Fig 4


Thus it follows that:

Concave mirrors produce two kinds of images or none at all, according to the distance of the object, as may be seen by looking at oneself in a concave mirror. If the mirror is placed nearer than its principal focus, then one sees an enlarged virtual image, which increases slightly in size as the concave mirror is made to recede; this image becomes confused and disappears as the principal focus of the mirror is reached; on moving the mirror still further away (that is beyond its focus) one obtains an enlarged inverted image, which diminishes as the mirror is still further withdrawn.


Reflection by a Convex Surface
Parallel rays falling on such a surface are reflected as divergent, hence never meet; but if the diverging rays thus, fortned are carried backwards by lines, then an imaginary image is formed which is called negative, and at a point called the principal focus(F).

Foci of convex mirrors are therefore virtual; and the image, whatever the position of the object, is always virtual, erect, and smaller than the object. The radius of the mirror is double the principal of the focus.

Fig 5


Refraction
Refraction by a Plane Surface

A ray of light passing through a transparent medium into another of a different density is refracted, unless the ray falls perpendicular to the surface separating the two media, when it continues its course without undergoing any refraction (Fig. 6, H K).

A ray is called incident before passing into the second medium, emergent after it has penetrated it.

A ray passing from a rarer to a denser medium is refracted towards the perpendicular; as shown in Fig. 6, the ray A B is refracted at B, towards the perpendicular P P.

Fig 6


In passing from the denser to the rafer medium the ray is refracted from the perpendicular, B D is refracted at C, from P P (Fig. 6).

Reflection accompanies refraction, the ray dividing itself at the point of incidence into a refracted portion B c and a reflected portion B E.

The amount of refraction is the same for any medium at the same obliquity, and is called the index of refraction; air is taken as the standard, and is called 1; the index of refraction of water is 1-3, that of glass 1-5. The diamond has almost the highest refractive
power of any transparent substance, and has an index of refraction of 2-4. The cornea has an index of refraction of 1-3 and the lens 1-4.

The refractive power of a transparent substance is not always in proportion to its density.

If the sides of the medium are parallel, then all rays except those perpendicular to the surface which pass through without altering their course, are refracted twice, as at B and C (Fig. 6), and continue in the same direction after passing through the medium as they had before entering it.

If the two sides of the refracting medium are not parallel, as in a prism, the rays cannot be perpendicular to more than one surface at a time. Therefore every ray falling on a prism must undergo refraction, and the deviation is always towards the base of the prism. The relative direction of the rays is unaltered
(Fig. 7).

Fig 7 Fig 8



If D M (Fig. 8) be a ray falling on a prism (A B C) at M, it is bent towards the base of the prism, assuming the direction M N ; on emergence it is again bent at N ; an observer placed at E would receive the ray as if it came from K ; the angle K H D formed by the two lines at H is called the angle of deviation, and is about half the size of the principal angle formed at A by the two sides of the prism.

Refraction by a Spherical Surface

Parallel rays passing through such a surface separating media of different density, do not continue parallel, but are refracted, so that they meet at a point called the principal focus.
If parallel rays E, D, E, fall on A B, a spherical surface separating the media M and N of which x is the denser; ray D, which strikes the surface of A B at right angles, passes through without refraction, and is called the principal axis; ray K will strike the surface at an angle, and will therefore be refracted towards the perpendicular C J, meeting the ray D at F ; so also witln ray E, and all rays parallel in medium M. The point F where these rays meet is the principal focus, and the distance between the principal focus and the curved surface is spoken of as the principal focal distance.

Fig 9


Rays proceeding from F will be parallel in M after passilig through the refracting surface. Rays parallel in medium N will focus at F', which is called the anterior focus

Had the rays in medium M been more or less divergent, they would focus on the principal axis at a greater distance than the principal focus, say at H; and conversely rays coming from H would focus at G; these two points arc then conjugate foci.

When the divergent rays focus at a point on the axis twice the distance of the principal focus, then its conjugate will be at an equal distance on the other side of the curved surface.

Fig 10


If rays proceed from a point O, nearer the surface than its principal focus, they will still he divergent after passing through A B, though less so than before, and will never meet by continuing these rays backwards they will meet at L, so that the conjugate focus of O will be at L, on the same side as the focus ; and the conjugate focus will in this case be spoken of as negative.


Refraction by Lenses

Refraction by lenses is somewhat more complicated. A lens is an optical contrivance usually made of glass, and consists of a refracting medium with two opposite surfaces, one or both of which may be segments of a sphere; they are then called spherical
lenses
, of which there are six varieties.


  • 1. Plano-convex, the segment of one sphere (Fig.
    11, B)

  • 2. Biconvex, segments of two spheres (Fig. 11, A)

  • 3. Converging concavo-convex, also called a converging meniscus

  • 4. Plano-concave

  • 5. Biconcave

  • 6. Diverging concavo convex, called also a diverging meniscus



Lenses may be looked upon as made up of a number of prisms with different refracting angles. Convex lenses, made up of prisms placed with their bases together and concave lenses, made up of prisms with their edges together.

A ray passing from a less refracting medium (as air) through a lens, is deviated towards the thickest; part, therefore the first three lenses, which are thickest at the centre, are called converging ; and the others, which are thickest at the borders, diverging.

A line passing through the centre of the lens, called the optical centre at right angles to the surfaces of the lens, is termed the principal axis, and any ray passing through that axis is not refracted. All other rays undergo more or less refraction.

Rays passing through the optical centre of it lens, but not through the principal axis, suffer slight deviation, but emerge in the same direction as they entered ; the deviation in thin lenses is so slight that they are usually assumed to pass through in a straight line, and are called secondary axes (Fig. 13).

Parallel rays falling on a bicolivex lens are rendered convergent; thus in Fig. 14 the rays A, B, C, strike the surface of the lens (L.) at the points D, E, F; the centre ray (B) falls on the lens at E perpendicular to its surface, and therefore passes through in a
6traight line ; it also emerges f ral the lens at right angles to its opposite surface, and so continues its course without without deviation; but the ray (A) strikes the surface of the lens obliquely at D, and as the ray is passing from one medium (air) to another (glass)
FIG. 13.
which is of greater density, it is bent towards the perpendicular of the surface of the lens, shown by the dotted line M K ; the ray after deviation passes through the lens, striking its opposite surface obliquely
Fig. 14.
at O, and as it leaves the lens, enters the rarer medium (air), being deflected from the perpendicular N O ; it is now directed to H,where it meets the central ray B H ; ray C, after undergoing similar refractions, meets the other rays at H, and so also all parallel rays falling on the biconvex lens (L).

Parallel rays, therefore, passing through a convex lens (L) are brought to a focus at a certain fixed point (A) beyond the lens; this point is called the principal focus, and the distance of this focus from the lens is called the focal length of the lens.
Rays from a luminous point placed at the principal focus (A) emerge as parallel after passing through the lens.

Divergent rays from a point (B) outside the princ1pal focus (F, Fig; 16) meet at a distance beyond (F') the principal focus on the other side of the lens (L), and if the distance of the luminous point (B) is equal to twice the focal length of the lens, the rays will focus at a point (C) the same distance on the opposite side of the lens; rays coming from C would also focus at n; they are therefore oalled conjugate foci, for we can indifferently replace the image (C) by the object (B) and the object (B) ty the image (C).

If the luminous point (D) be between the lens and the principal focus (F), then the rays will issue from the lens divergent, though less so than before entering it; and if we prolong them backwards they will meet at a point (H) further from the lens than the
point D ; H will therefore be the virtual focus of D, and
the conjugate focus of D may be spoken of as negative.

Biconvex lenses have therefore two principal foci, F and F', one on either side, at an equal distance from the centre.

In ordinary lenses, and those in which the radii of the two surfaces are nearly equal, the principal focus closely coincides with the centre of curvature.

We have assumed the luminous point to be situated on the principal agi-s, supposing, however, it be to one side of it as at E (Fig. 17), then the line (E F) passing through the optical centre (C) of the lens (L) is a secondary axis, and the focus of the point E will be found somewhere on this line, say at F, so that what has been said respecting the focus of a luminous point on the principal axis (A B) is equally true for points on a secondary axis, provided always that the inclination of this secondary axis is not too great, when the focus would become imperfect on account of the spherical aberration which would be produced.

FiG. 17.

In biconcave lenses the foci are always virtual whatever the distance of the object. Rays of light parallel to the axis diverge after refraction, and if their direction be continued backward they will meet at a point termed the principal focus (Fig. 18, F).

Fig. 19 shows the refraction of parallel rays by a biconcave lens (L) ; the centre ray B strikes the lens at E perpendicular to its surface, passing through without refraction, and as it emerges from the opposite side of the lens perpendicular to its surface, it continues in a straight line ; the ray A strikes the lens
obliquely at D and is refracted towards the perpendicular, shown by the dotted line G H ; the ray after
FIG. 19.
deviation passes through the lens to K, where, on entering the medium of less density obliquely, it is refracted from the perpendicular O P, in the direction K M ; the same takes place with ray C at F and N ; so also with all intermediate parallel rays.

Formation of Images. To illustrate the formation of images the following simple experiment may be carried out: Place on one side of a screen having a small perforation, a candle, and on the other side of the screen a sheet of white cardboard at some distance from it to receive the image formed; rays diverge from the candle in all directions, most of those falling on the screen are intercepted by it, but some few rays pass through the petforation and form an image of the candle on the cardboard, the image being inverted because the rays cross each other at the orifice.

It can further be shown that when the candle and cardboard are equally distant from the perforated screen, the candle flame and its image will be of the same size. If the cardboard be moved further from the screen the image is enlarged, if it be moved nearer it is diminished; if we make a dozen more perforations in the screen, a dozen more images will be formed on the cardboard, if a hundred then a hundred ; but if the apertures come so close together that the images overlap, then instead of so many distinct images we get a general illumination of the cardboard.

The image of an object is the collection of the foci of its several points; the images formed by lenses are, as in the case of the foci, real or virtual. Images formed, therefore, by convex Ienses may be real or virtual.

In Fig. 21, let A B be a candle. situated at an infinite distance; from. the extremities of A B draw two lines passing through the optical centre (C) of a bicorlvex lens, then the image of A will be formed somewhere on the line A c (termed a secondary axis),
say at a ; the image of B will be formed on the line B c b, say at b; therefore b a is a small inverted image of the candle A B, formed at the principle focus of the convex lens. Had the candle been placed at twice the focal distance of the lens, then its inverted image would be formed at a corresponding point on the opposite site of the lense, and would be of the same size as the subject.

If the candle be at the principal focus (F), then the image is at an infinite distance, the rays after refraction being parallel.

If, however, the candle (A B) be nearer the lens than the focus, then the rays which diverge from the candle will, after passing through the conves lens, be still divergent, so that no image is formed; an eye placed at E would receive the rays from A B as is they came from a b; a b is therefore a virtual image of A B, erect
and larger than the object, and formed on the same side of the lens as the object.

Images formed by biconcave lenses are always virtual, erect, and smaller than the object. Let A B a candle, and F the principal focus of a biconcave lens; draw from A B two lines through C, the optical centre of the lens, and lines also from A and B parallel to the
axis; after passing through the lens they diverge and have the appearance of coming from a b, which is therefore the virtual image of A B. A real image can be projected on to a screen, but a
virtual one can only be seen by looking through the
lens.