Optics

Optics

So far, the wall, met by rays of light, was to reflect the light in all directions uniformly - diffusively. Imagine that the floor is a wall of a room (Fig. 624) into which rays of the Sun fall through a small opening. The bundle of rays meets the floor at B. From whatever direction an observer looks towards B, he sees the spot lit up. This is true as long the floor reflects the light in all directions, like every area which is rough and appears to be mat and dull. However, if there is at B a mercury surface at rest, the observer sees the spot B effectively without a shine. Now he sees at C (Fig. 625) a spot of light, because the wall is met there by a bundle of rays coming from B. If the observer turns his eye in this direction, he sees at B not the mercury surface, but the source of the light, which sends the rays into the room - the sun.

You call the mercury surface a mirror, AB the incident, BC the reflected ray, BN the incidence perpendicular, ABN the angle of incidence, NBC the angle of reflection, the plane containing the incident ray and the perpendicular is the plane of incidence - in Fig. 625, the plane of the drawing.

The relationship between the incident and reflected rays is: Also the reflected ray lies in the plane of incidence, on the opposite side of the perpendicular and forms with it the same angle as the incident ray, that is, ABN = NBC.

This basic law arises from experience, but can also be derived from the theory of waves (first by Huygens, then by Fresnel). Its validity is best verified by astronomical observations: The horizontal surface of a pond presents the mirror image of the sky with its stars. If you observe a definite star, you see it once in the sky, a second time deep down in the pool. Look at it with a telescope and direct it first upwards, the second time downwards. If the telescope's axis F forms in each position the diameter of a vertically inclined circle, subdivided into degrees (Fig. 626), you can confirm that a = a', that is, the axis forms both times the same angle with the horizontal HH, which is parallel to the surface of the pond. Since the rays come from an infnitely far away point, they are parallel, whence:

a = b ' (angles with pairwise parallel and equally directed legs),
a ' = b ' (for the same reason),
whence a = b ' (since a = a' ) and,
finally, i = i' (each angle is equally far away from 90º, the one by b, the other by b '.

We have only talked about a single incident and a single reflected ray, but you should consider them to be representatives of bundles of parallel rays, since there do not exist individual rays.

Mirror image Virtual, real image

We know that a mirror can generate images. The deception, caused by a mirror image, can be so true that the image can hardly be distinguished from the reflected object. How does this deception arise? At the mirror (Fig. 627), you must extend the rays backwards from the mirror, in order to reach their point of intersection (mirror point); in the case of optical images, which arise through a photographic lens, they actually intersect (Fig. 694). Hence you can catch in the second case the image by a screen (mat glass), while you cannot do this in the first case; you can only observe subjectively the image. The second kind of images is said to be real, the first inaccessible.

Moreover, in the case of a lens, the image may also be inaccessible (a magnifying glass, spectacles can only serve for looking through and not for making the image real). In the case of an accessible image, the rays converge behind the magnifying glass, in the case of a real image, they diverge. If you design a real image, you can place between the lens and the image an optical instrument, which then receives a bundle of rays converging towards one point - a non-real, virtual object. In contrast, a real object emits a diverging bundle of rays. At the back of the image point, real and inaccessible images give the impression of real objects.

A shiny point emits rays; if a sufficient number reaches your eye, you see it. By experience, we have become so conscious of the fact that light travels away from its source along straight lines, that our eyes always transpose the point, which it considers to be the source of the light, along the ray away from it - along a straight line - whence it considers L₁' and L₂' to be sources of light (Fig. 627). This deception arises just as in the cases of an echo, when the ear constructs a source of sound, and of the reflection of water waves (Fig. 306), when the eye constructs the centre of the reflected system of waves. In reality, there does not exist a new (real) system of waves; however, the eye gets the impression of their existence; it is a virtual one!

The backwards extensions of the reflected rays (Fig. 628) must pass through a point which is common to all of them, for they have the same geometrical relationship between them and to the line HH, which the incident rays have to each other and HH; indeed, all of them share the point O. The symmetry of the incident rays and their backward extensions to HH also yield that the distances of the points of intersection 0, 1, 2, 3 from the object O and the image B are equal. - Also the ray O0, which is perpendicular and reflected, must pass through B, whence also O0 = B0, that is, the image point lies on the opposite side of the mirror and equally far away from it as the object point.

These facts enable you to construct for an object its mirror image You draw from every reflectable point of an object the perpendicular to the mirror and extend it beyond by the distance of that point from the mirror (Fig. 629). However, this construction of the image does not yet guarantee its visibility. Visibility does not demand that certain rays, for example, those used in the construction, meet the mirror and reach on reflection the eye; any rays will do, but they must really meet the mirror and then really meet the eye. This is demonstrated by Fig. 630 at the wall W, which almost extends to the mirror, so that it shields many of the rays from O from the mirror and many reflected rays from the eye. Only the cone of rays BOc makes the image visible: Oc and OB are the bordering rays and only as long as the eye lies in the cone BB'c, it will see B'. Rays such as the ray Ob do not reach the mirror and those like Oa, although reflected, do not reach the eye. Hence, the effective rays form only one part of the total number of rays starting from O; they lie within a limited space. This process is called ray delimitation. In optical mapping equipment such as magnifying glasses, microscopes, telescopes, photographic lenses, you delimit by diaphragms a section of the rays, coming from individual points of an object, and allow these to partake in the mapping; you improve, indeed only make possible thereby the formation of optical images.

Angled Mirror

If rays reflected by one mirror encounter a second mirror, they are again reflected; if you arrange that they then again meet the first mirror, then again the second, etc., you see many images, which differ in numbers and positions with respect to each other, depending on the size of the angle which the mirrors form with each other (Fig. 631). The 90º-angled mirror has a special role. As we know already, an ordinary mirror interchanges left and right. For example, standing in front of a mirror, if you incline your head towards the right shoulder, the mirror image bends its head towards its left shoulder. It is different in front of a 90º-angled mirror; if you bend now your head towards the right shoulder, the mirror image also does this. What for you lies towards the right, also lies there for the central mirror image, intersected by the common edge of the mirrors (mirror axis) towards the right (Application in the prism telescope of Zeiss ). Fig. 632 explains: Let the plane of the figure be perpendicular to the edge of the angled mirror (mirror axis), A be its intersection with the edge, AB and AC their intersections with the mirrors, the plane of the drawing also be the plane of incidence (and also the plane of reflection), O a point in it, which emits rays into the angled mirror. Its reflection in AB generates the image O₁, the reflection in AC takes it from O₁to O₂, whence the double reflection turns it (with its distance AB from the edge) by the angle OAO₂ about the mirror axis. The angle of rotation OAO₂ is twice the mirror angle BAC, because

OAO₁ = 2O₁AB
		O₁AO₂ - OAO₁ = OAO₂ = 2(O₁AC -O₁AB)= 2CAB
O₁AO₂ = 2O₁AC

Reflected in an angled mirror, a twice reflected point describes an arc which is twice the angle of the mirror,whence the 90º-angled mirror turns every twice reflected point by 180º so that in Fig. 632 becomes in the mirror image.

Application of mirror in measurement of angles

There exist many applications of mirrors to physical measurements, for example, for the measurement of angles. In Crystallography, a determination of the magnitude of an angle is the purpose of the measurement, but very frequently italso yields knowledge of other physical quantities. Thus, in order to measure the angle a between two reflecting faces of a prism at c (Fig. 633), you would proceed as follows: In Fig. 635 and Fig. 636, T is a circular table (seen vertically from above), its circumference is subdivided into degrees. You can turn the table about its vertical axis through its centre. You place the object approximately at the centre of the table, so that the edge EF is vertical; if you look across the table along a given direction RR through the centre of the table and perpendicularly to the table's axis of rotation and turn the table until you see your eye reflected in the face cb of the prism, you know that a ray from your eye along RR to cb returns to it. The side cb is then perpendicular to the direction of sight, that is, the angle W is a right one (Fig. 634). If you now continue to turn the table to B until, always looking along RR, you see your eye reflected in the side ca, the angle V in Fig. 634 is a right one. You read at the edge of the table the angle, by which you have turned the table, in order to turn the object from the first to the second position,. The angle at c, which was to be measured, complements it (B in Fig. 634) to 180º.

Fig. 636 shows the the spectrometer of Ernst Karl Abbe 1840-1905; it employs this method. A incandescent straight line as source of light in the specially designed telescopic ocular throws light through the telescope F on to the reflecting area which you inspect simultaneously through F. You let the light enter through an opening at P, in order to generate the shining line. You see in the field of vision the source of light and its image reflected by the inspected area and recognize distinctly when during the rotation of the small table T the source of light and its image coincide, indicating that the inspected area is perpendicular to the direction of view. The handling of the spectrometer during measurement of angles follows then from above.

Sextant (Hadley ?-1744)

The sextant is an instrument for the measurement of angles; it is important for navigation. In essence, it is an angled mirror, the angle of which can be changed and thereby adjusted to the angle to be measured. For example, in order to determine (in connection with other measurements) the geographic latitude and longitude of the location of a ship, you measure from the ship the arc distance of the sun from the horizon (sea level). This is the angle between the two directions (lines of view) from the eye, say, to the lowest point R of the sun's rim and the point L which lies below it at the horizon.The angle between the arrows R and L (coming from these points) is measured with the sextant (Fig. 637). S and s are the faces of the angled mirror; s is fixed, S can be rotated with the lever A, so that the angle between s and S change; you read its magnitude on the graduated arc which, as a rule, is one sixth of a circle, whence originates the name sextant. You have to imagine that your eye, looking through the telescope, is at the vertex of the angle a. The rays coming from L pass directly via the mirror s into the telescope, those coming from R after reflection in S and s, that is, after double reflection. You turn the mirror S for the measurement of the angle by A along the arc until the point R coincides with the point L, that is, the ray R after reflection in S and s has the same direction as L. The angle a between R and L is then twice as large as the angle b between the two mirrors.You read off the scale twice times the angle of the mirror.

Measurement with mirror and scale (Poggendorf 1825)

One of the most important methods of physical measurement employs the law of mirrors for the measurement of very small angles by which a rotatable body deviates from its initial position. The task is as follows: The vertically suspended cylinder in Fig. 638 can rotate about the axis DB; let it be turned from its position of rest AA by the small angle a and then held fixed. How large is the angle? You attach to the body a small mirror S so that its plane is parallel to the axis of rotation; it will then take part in the rotation of the body; its position before the rotation is AA and after the rotation EE, so that ECA is the angle a to be measured. For its measurement, you place a mm ruler about 2 - 4 m in front of the mirror A, parallel to the starting position AA (Fig. 639), so that the zero N of the scale, with subdivisions in both directions, is perpendicularly opposite to the mirror. If you look in the perpendicular direction RR just below the zero of the scale towards the mirror (through the telescope), you see always that point of the scale in the mirror, the rays of which meet the mirror in such a way that they are reflected in the direction RR. In the position of rest AA, you see therefore the zero N (with the number 0), in the final position EE, the point P (with a number on the scale). While the mirror rotates, your eye gets the impression that the scale moves along from N to P. The length NP is the objective of the measurement; you read the number at P, measure NC with a ruler and find the angle PCN from the formula NP/NC = tan PCN. The angle PCN equals 2a, because the perpendicular pC to the mirror position EE bisects the angle PCN (because pCN = pCP, according to the law of reflection); moreover, pCN = a (because pCN and ACE complement the same angle ECN to 90º), whence: NP/NC = tan 2a, that is, a = ½artan NP/NC.

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