Optics

L5 Refraction and colour dispersion of light

Refraction

Hitherto light was to meet a wall which impedes its progress; the wall was not to absorb the light, but to reflect it. We will now assume that the wall lets all of the light pass, that is, that is is perfectly transparent. (In reality, this ideal state cannot be achieved, but this is not important here.) Moreover, let the wall be isotropic - the reason will become apparent in the discussion of double refraction .

What happens when light reaches a wall? Let the wall be a very thick, vertically placed glass plate, the point source of light L on the one side, an observer B on the other side. Let the observer hit L with a bullet. If he aims his gun at where he sees the light, the bullet will fly past above it (Fig. 647) - this is a fact! The reason is: The gun is aimed in the direction of the arrow B, because he seeks the point, from which the light comes along the backward extension of the rays, which reach his eye. However, this line passes above L . The figure shows the reason: As the ray from L encounters the glass - enters the glass from the air - it changes its direction. It continues along the new direction while inside the glass, but changes it again, as it leaves the glass, that is, as it enters the air. In the present case, while the ray proceeds in a direction parallel to its initial path, its new path is not a prolongation of the initial path; it thus misleads the observer regarding the location of the light. You describe this process in the following words: The ray of light is refracted, as it enters the glass from the air, and then is again refracted as it leaves the glass. The line, starting in Fig. 647 at L , demonstrates the ray's refraction. (However, it is assumed here that the line of view, as in Fig. 647, is inclined to the glass. If it is perpendicular to it, the ray is not transposed.)

For the same reason explained in connection with Fig. 647, a body on the bottom of a container filled with water appears to you to lie higher up than it really is, the bottom of the vessel seems to be higher up, the water less deep than they are; points on a bar standing perpendicularly in the water seem to lie higher up, that is, the bar seems to be shorter; points of a bar lying at an angle in water (Fig. 648) seem to be raised, that is, the bar seems not to be straight but angled, etc.

A similar delusion is due to atmospheric refraction of rays. In order to reach Earth from a star, light must enter from vacuum into its atmosphere and pass downwards through air of increasing density. During the transition from one layer into a denser one, light is refracted towards the incident perpendicular (Fig. 649). As a consequence, a star seems to be higher over the horizon than it really is. - You can convince yourself of the fact that the atmospheric air due to its varying density deceives an observer regarding the location of an object by looking across a flame (a Bunsen burner or a lamp with an open flame). You see then how objects glitter: The air, heated by the flame, has less density, rises and mixes thereby differently dense layers. The rays of light which must pass through these layers in order to reach your eye change all along their direction. As a result, the objects move quickly to and fro, that is, they glitter. The superposition of differently refracting layers of air also explains the Fata Morgana, which displays to the wanderer in the desert far away objects as if they are nearby.

Dispersion

Another phenomenon is closely linked to refraction of light. If the planes of a transparent body in between an observer and a source of light are not parallel, but inclined to each other (Fig. 650), the observer sees the source of light spread out with a coloured border: Colour arises without a presence of a colouring agent.

The experiment of Fig. 651 explains the process: A horizontally directed ray of light enters the space through a small circular opening. In order to reach the opposite wall, it must pass through the glass prism P. Let the refracting edge of the prism be horizontal and perpendicular to the incident ray bundle. We know already that the prism refracts the light and expect to see the circular spot of light, which we would observe at a if there were no prism, at some other spot. However, we find instead a vertical, longish strip rv with semi-circular ends at r and v and - it is the main point - it is red at r and violet at v. Its centre is almost white, but transits through gradual changes of colour to red and violet, respectively (Newton 1666).

We interpret this phenomenon as follows: The white light of the Sun consists of a large number of coloured kinds of light. According to the old, somewhat arbitrary decomposition, they are: Red, orange, yellow, green, blue, indigo,violet. These kinds of light are refracted differently strongly. Red light alone would form a red circle of light at r, violet a violet one at v, orange, yellow, etc. spots in their colours in between. At the centre between r and v, if the opening is large enough, the spots coincide and give tby their combination the impression of white, formed by the source of light through the kinds of light it contains. However, colour appears at the borders, where they only superimpose partly or not at all like at the ends r and v,. - The red (violet) spot lies closest (furthest away) from the point a, that is, it is least (most) deflected from it, whence you say that the red (violet) rays are least (strongest) refractable.

Spectrum

This process is called colour dispersion of light. The band of colour is called the spectrum* of the source of light. The spectrum, shown on the left hand side of Fig. 651, is not clean, because the individual spots of colour are partly superimposed. If you ensure that they only appear close to one another, for example, by making them so small that at most two immediate neighbours interfere with the furthest away rims, the spectrum becomes pure and forms a graduated band of colours ranging from red to violet (Fig. 652). Then the wave length, by which the different colours differ sharply physically, varies gradually. For the extreme violet, which we still just sense (Helmholtz), the wave length is l = 0.000396 mm, for the extreme red l = 0.000760 mm. In order to define a colour by the words blue, yellow-green, etc., you must state its wave length.

* The word spectrum, which Newton employs in his English text and which has become the technical term for the band of colours, means a (non-corpuscular) phenomenon; it appeared already in classical Latin and was used in the 17th Century for parhelions as well as for ghosts.

For example, you can generate a spectral colour of definite wave length by screening off from the spectrum of while light (which contains all colours) all components except for the required wave length. Discharge valves or making certain substances glow also yield colours of exactly defined wave lengths. For example, if you place cooking salt into a not glowing flame of a Bunsen burner, you obtain yellow light with the wave length 0.000,589 mm, characteristic for sodium (measurement of wave lengths).

You obtain a much purer spectrum by letting the light enter the prism through a very narrow slot (Wollaston) which is parallel to the refracting edge of the prism, if you make sure that the rays of the incident bundle of light are parallel to each other and the distance between the prism and the wall is very large (Fig. 651 right). In fact, if the rays entering the prism are parallel, the rays of the same colour leaving the prism are also parallel - the green ones parallel, etc. Hence rays of different colours mix less readily at their exit than if they exit as divergent rays. - You obtain a completely pure spectrum by letting the rays coming from the prism pass through an achromatic convex lens. The separately exiting parallel rays of a bundle remain then also separate, but every bundle on its own forms on the wall like through a photographic lens a sharply split image corresponding to its colour. The individual bands lie then close side by side.

Fraunhofer lines (1817*)

*The first observation of the dark lines in the solar spectrum was made by Wollaston.

The solar spectrum contains thousands of extremely fine spaces, which cross it as black, straight lines (Fraunhofer). (Their origin is the subject of spectrum analysis. ) Each of them corresponds to a definite simple colour which (apparently) is not contained in the solar spectrum. Hence you denote the location of each Fraunhofer line by the wave length of the colour, which ought to be seen there. The wave lengths, to which the pronounced lines A - H in Fig. 652 correspond are according to Helmholtz:

In order to avoid the many zeros in the wave lengths, you set 0.0001 mm = m, whence 0.000001 mm = mm. Moreover, one also calls 0.000,000,1 mm, that is, mm/10 or mm^-7, an Ångström unit (1 Å.-E)*. Hence there corresponds to the line D in Fig. 652 the wave length l = 0.000589 = 0.589m = 589 mm = 5890Å.-E. and the wave number 10⁸/5890 cm^-1. As a rule, you write l = 5890. The large significance of the Fraunhofer lines is that they designate certain locations in the spectrum and thus serve as points of reference during examination of the refraction ratios of a substance. In order to obtain unique statements, we seek the refraction ratio of a substance for light which would correspond to the lines A, B, ···. The terminology green and yellow is not unique, since there exists green and yellow light with different refractivities.

* In the field of X-rays, in order to denote wave lengths, which are 1000th shorter, you employ, following Siegbahn, as length unit mm^-10 = cm^-11 = 1 X. For 3,351 Å.-E you write 3351 X-E.

Method of crossed prisms (Newton 1672)

The individual colours of a completely pure spectrum cannot be further decomposed: If you let the rays of a given colour of the spectrum, for example, the top red of Fig. 651, fall through a hole in a wall into a second prism, the colour - red - is again deflected by the prism, but not further decomposed - it has a single colour; you say also that it is simple or homogeneous. If you place the refracting edge of the second prism parallel to that of the first prism, it diverts the colour vertically, that is, parallel to the length direction of the prism (downwards or upwards depending of whether the edge in Fig. 651 lies above or below). However, if you place the second prism upright, its refracting edge vertical, so that is runs across, it diverts the colour sidewards; this is true for every colour of the spectrum. In this manner, you obtain a new spectrum - exactly as wide as the first - a proof that the second prism cannot decompose further the colours of the first spectrum.

Crossed prisms have a principal role in the examination of anomalous dispersion . Fig. 653 shows the Kron-Flint-prism B (with straight transparency in order to bring the figure into a single plane), its refracting edge K₁ placed vertically, the second prism (P), its refracting edge K₂ placed horizontally. The light comes through a slit (in the collimator tube C) from the right hand side; imagine the observing eye (at the telescope F) to be on the left hand side. Without K₂, the observer would see the spectrum in the normal form (d). Looking through the second prism, he sees the coloured split images, diverted more or less upwards according to their refractibility (through the telescope downwards!), so that the spectrum (c) extends inclined through the field of vision.

Complementary colours

All the simple colours are contained in the solar spectrum (spectral colours). If you mix them again - in the ratio in which they are contained in the Sun's light - the mixture is white. However, if you omit just a single colour A, you obtain again a colour B, and only on adding a colour B you obtain again white, that is, A complements B into white. You say of both the colours, which mixed in a definite ratio yield white, that they are complementary colours. - Every single spectral colour is also a complementary colour to that colour, which a combination with the other spectral colours yield. However, there exist also certain single spectral colours which already together with another colour yield white. According to Helmholtz, such complementary colours are:

The green of the spectrum has the single composite complementary colour purple.

Note that we are here only concerned with kinds of light, not with coloured substances. A mixture of indigo-blue and yellow painter's paint does not yield white, but green. The colour substances are coloured because the white light which falls on them is partly absorbed, partly reflected. - We will not deal here in detail with colour sensation and theory; a superficial treatment would serve no purpose, a detailed one is the task of Physiology, not of Physics.

Rainbow

A solar spectrum of cosmic dimension is the rainbow. It forms a circular arc out of a spectral band of colours - inside blue, outside red; it arises by double refraction and in between the two single reflections of the Sun's rays in rain drops (and refraction interference, which, for example, explains that the sequence of colours is different in almost every rain bow) and depends on the observer having the Sun in his back and the rain cloud in front (Fig. 654).. In as far as one only refers to refraction and reflection (Antoniusn de Dominis, arch bishop of Spalato, 1611; Descartes, Newton), as is not sufficient for a complete explanation (Sir George B.Airy 1801-1892), the geometry is as follows: You see at s the Sun's rays, which on entry to the rain drops are refracted, in the drop reflected and on exit again refracted. Only such rays contribute to the formation of the rainbow, but also only a fraction of them, because the rays, which fall parallel to each other on the various points of the surface of the rain drop, are on their exit no longer parallel; the not parallel ones or at least not nearly parallel ones do not act on the eye. Only a certain group of rays enters drops which exit it again parallel. A solar ray which enters a drop is deflected from its initial track by the refraction on entry, reflection in the drop and refraction on exit. The final deflection is measured by the angle between the entry and its direction (Fig. 655). In the direction of that exiting ray, which is least deflected (minimal ray), the brightness of the emerging light is largest, because also the rays emerging in the neighbourhood of this ray are nearly parallel to it and hence effective for the eye (below the minimum ray, no rays whatever leave the drop; those which exit above it form angles of considerable magnitude and are of no consequence for the eye). The minimum of the deflection of red is about 137º 58'. Hence, if you convert in Fig. 654 the angle at O into 42º 2', you obtain the direction in which the brightest red radiates. For the violet deflection, the minimum is 139º 43', the corresponding angle at O for the active rays of violet is therefore 40º17'. For the other colours, the most favourable values of entry and exit lie in between those for red and violet.

Due to the Sun's large distance, all its incident rays can be considered to be parallel. If you draw parallel to them through the eye the line OP, then red (violet) light must fall into the eye along every line which forms with OP an angle of 42º 2' (40º 17'); these lines form a conical surface, the vortex of which is in the eye and which intersects the sky in the red (violet) circle of the rainbow. This explains the circular shape of the rain bow and the sequence of the spectral colours from red to violet (from above to below). (The colours are not sharply divided; they interfere with each other and some of them are hard to detect, because the Sun has a diameter of 33' and emits rays from each point, whence there arises a sequence of rainbows which are superimposed on each other and make it less sharp.) At times, you see outside the rainbow a second one, which is formed by double refraction and reflection in drops above and which has much weaker light. The sequence of the colours in it is inverted compared with that in the first rainbow.

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