Refraction of light by planes
Law of refraction of light
Refraction changes with the colour of the refracted light. In order to simplify the following discussion, we will employ first of all only mono-chromatic light, for example, yellow light which is generated by cooking salt in the colourless flame of a Bunsen burner (Fraunhofer's line D), and explain the terminology and notation of refraction.
If you let the bundle of light AB (Fig. 656) pass through water, made turbid by milk, - the drops of milk display the track of the light like dust in air - the path of the bundle BC in the water is not an extension of the bundle AB in the air, but it forms an angle with it. (One part of the light - BR - is reflected, but we will disregard this fact.) You call this process refraction, the surface FF (Fig. 657) separating the media air and water the refracting surface, the ray AB the incident ray, BC the refracted ray, the perpendicular BN the perpendicular of incidence, r the angle of refraction, the plane in which lie the incidence perpendicular and the incident ray the plane of incidence (here the plane of the drawing).
You relate the direction of the ray to the perpendicular of incidence (like in the case of reflection). The relationship between the incident and refracted rays is the subject of the law of Snellius (about 1630). Also the refracted ray lies in the plane of incidence, and, indeed, on the opposite side of the perpendicular of incidence; the sin of the angle of incidence divided by the sin of the corresponding angle of refraction is a fixed number n, the magnitude of which depends on the nature of the adjoining substances (here air and water) and the colour of the light:
nab = sin i/ sin r.
The constant n does not depend on the magnitudes of the
two angles i and r,
and a and b denote the two substances through
which the light passes. During transition of yellow light (Fraunhofer-line D)
from air (a) into water (b): nab = 4/3,
sin (angle of incidence in air)/sin (angle of refraction angle in water) = 1.33, instead of which you write nD = 1.33.
In order to understand this equation, imagine that the intersection of the incidence plane through the vessel filled with water is circular (Fig. 568), the vessel is half full and the experiment is such that a ray incident from the left hand side always meets the centre; if you now erect the perpendicular of incidence NN, then, for example, if r is is the radius of the circle, e1E1/r : b1B1/r = 4/3 and similar for e2E2 and b2B2, etc. Hence the incident point E1 or E2 lies from the perpendicular NN 4/3 times as far as the corresponding location B1 or B2, at which the refracted ray meets the vessel in the water.
The number 4/3 is called the refractive index (-exponent, -quotient, -coefficient, -number) of air with respect to water. The following table presents the refraction index nD of air with respect to the stated substances:
|glass (flint)||1.54-1.8||carbon disulphide||1.6204|
|alcohol||1.3617||air (free from CO2)||1.000293|
Every colour of light has a different n, whence you must state for every n the colour involved (Maxwell's relation). Thus, nD denotes the refractive index for the yellow, corresponding to the Fraunhofer-line D.
The basic law of refraction comes from experiments, but it can also be derived from the wave theory of light. The law is confirmed by measurement of the refraction ratios at different angles of incidence, and especially by the agreement of the computed values and optical equipment, designed by means of it.
Reversibility of ray paths
Moreover, experiments show: If the light passes during its refraction the two substances in reverse order, that is, it passes (Fig. 659) first through water and then through air, and the ray travels in water (now as incident ray) along the same path from B to C, which previously it had to travel as refracted ray from C to B, it travels as refracted ray in air from C to E which it previously travelled as incident ray from E to C. If the refracted ray CB in Fig. 660 encounters a mirror at a right angle at B, it returns to itself and along the entire way along which it came. This fact is referred to as the principle of the reversibility of the path of a ray.
During an inversion, the angle r becomes the angle of incidence and the angle i the angle of refraction, whence sin r/sin i = nb,a, where nb,a denotes the refractive ratio during the passage of light from water into air. However, from sin i/sin r = na,b follows sin r/sin i = 1/na,b, whence nb,a = 1/na,b, that is, if nair,water = 4/3, then nwater,air = 3/4.
These number apply for the transition of light from air into water, from air into glass, etc.; however, they also yield the refractive index between glass and water. In fact, experiments tell us: If na,c is the refractive index from the substance a to c and nb,c that from b to the same c, then na,c/nb,c = na,b, that is, it is equal to the refractive index from a to b. If a refers to water, b to glass, c to air, then
= nwater,glass or also (since na,c
nair,glas/nair,water = nwater,glass
For glass with n = 1.5, you have nair,glass = 3/2, nair,water = 4/3, whence nwater,glass=3/2·3/4=9/8, a result, confirmed by measurements.
The relationship na,c/nb,c = na,b or, what is the same thing, nc,b/nc,a =nab, reduces the number of measurements which would have to be made to determine the refractive ratios of substances in pairs: You measure the refraction of all substances with respect to one - say, air, that is, you measure the refraction into all substances from air and then compute all the other indices.
The refraction index for the transition from one substance into another is called the relative index; the absolute values apply to the transition of light from empty space into a substance. If we replace the substance c by vacuum and denote the absolute refractive indices from a to b by na and nb, then na/nb = na,b. The equation sin i/sin r = na,b then becomes na·sin i=nb·sin r.
Let the wall (out of glass) which light encounters as it spreads be perfectly transparent. This means: The light, as it arrives at the surface of the wall, passes through the interface between the glass and air without being weakened. However, this never happens, since one part of the light is always reflected, even at the surface of so transparent substances as water and glass, into the substance from which it has arrived. The surface of water reflects, so does every pane of glass like the mirrors of shop windows. However, the reflected light is always only a small part of the incident light, while it becomes the larger, the more inclined the rays are to the surface they encounter. Such reflected images have therefore always very weak light. Their formation is always accompanied by refraction, which is the principal part of the incident light passing through the interface.
Note: During this
process, the light comes from air and enters water or glass. However, it is
different when it comes from
water or glass and enters air. When a ray of light comes
from air and enters water or glass, the angle of refraction is always smaller than the angle of incidence;
however large is the angle of incidence (Fig. 661, left image,
between 0º and 90º), there always is added an angle of
refraction. However, it is different when the light comes from water or glass and enters air.
The right image in Fig.661 explains: The angle of incidence r
in water is then largest when the angle of refraction in air
is a right one, because this is the largest angle, which the ray
can form with the perpendicular NN in the semi-circle of
air. This angle r is the limiting angle. The rays in Fig. 662, which form a still larger angle with the normal, for
example, g, cannot
at all exit into the air, they are reflected into the water, that is, they are
reflected at the interface of water and air. This process is
called total reflection, because all rays
are reflected, as you conclude from the fact that the reflected
light has the same strength as the incident light. In brief: If
light passes from a more
substance in to a more
weakly refracting one,
that is, into a substance, in which the ray is refracted away
from the perpendicular and exceeds the angle of incidence by a
certain value, the light is reflected altogether back into the
more strongly refracting substance. - How large is the limiting angle? If in
air the angle of
incidence i is like in Fig. 661 (left image) a right
one, then sin 90º/sinr = n, and since
sin 90º = 1, sin r = 1/n.
The angle r is the limiting angle: Imagine the sequence in which the ray passes through the substances in the reverse order (Fig. 661 right image), you see then that r is the angle of incidence, to which is added an angle of refraction: The emitting ray touches the surface. For water, n = 1.33, whence the limiting angle is that angle the sin of which is 3/4, that is, the angle 48º 36' 25; for one kind of glass (light crown glass ) with n = 1.50, the limiting angle is 41º 48' 37|.
You can observe total reflection at a water surface which you view - as in an aquarium - from down below, but also by looking at it in a glass of water from below. You cannot see through the water surface and get the impression that it is a perfect mirror.
The total reflection of light explains how certain objects which as a rule are shiny and at the same time transparent, become under certain conditions opque and dull, for example, powdered glass and ice, snow, foam (irrespectively whether it is made out of coloured or colourless substances). Powdered glass is, strictly speaking, a mixture of glass and air. It is dull for the same reason as, for example, polished silver is, when it is powdered: It turns its reflecting faces in all possible directions, that is, throws light in all directions, that is, it is diffusive. It is opaque due to the total reflection of light in the mixture of glass and air; light cannot pass through the mixture. Powdered glass becomes transparent, if you cover it with oil, which has approximately the same refractive ratio as glass.
Totally reflecting and refracting surfaces as substitute for metal mirrors
The optical industry employs totally reflecting, refracting planes as substitutes for metal mirrors, because they reflect much better. For example, Fig. 663 demonstrates its application at the hypotenuse of the right-angled prism A or Fig. 740 at short sides of each of the two right-angled prisms. You employ reflective prisms, in order to change systematically the direction of the rays of light. Fig. 663 displays this for the ray s, Fig. 740 for the ray with an arrow. Occasionally, you employ several prisms for this purpose (Fig. 663) or you manufacture objects out of glass which correspond to such action (Fig. 664).The first is done in the camera lucinda of Wollaston 1809, by means of which you can draw natural objects and which you can combine with a microscope.Fig. 663 shows the prism pair of the camera lucinda of Zeiss. The rays coming from the cusp reach your eye only by the round about path through the camera lucinda, and indeed out of the same final direction, from which come also the rays from the microscope; hence the eye projects the cusp on to the image seen in the microscope. You view through the camera the paper, lying next to the microscope, and the pencil on the image in the microscope, with which you can draw the outlines. The rays from the microscope do not pass through the prism, but pass its rim on their way to your eye, in order not to be refracted by the prism.
Fig. 664 shows a totally reflecting glass object, which corresponds to a combination of three prisms: The equilateral, right-angled prism ABC, bounded by two other right-angled prisms each of which has a refracting angle of 30º. The first serves total reflection, the two others prisms the refraction of the incident and exiting rays. If the incident ray meets AF at such an angle of incidence that it after refraction passes through AB perpendicularly - then you must have sina /sin 30º = n, that is a = n/2 - then it also passes - reflected in AC at 90º - also perpendicularly BC and exits at the angle a from FE (since the two 30º prisms lie symmetrically to the two halves of the ray). In each of the two 30º prisms, the ray passes the prism in the minimum orientation (cf. below). This is important for the application of a glass body in a spectral instrument. Every differently coloured light has another refractive index n, corresponding to the relation sin a = n/2, that is, another anglea . You change a by turning the glass body and thereby directing AF differently with respect to the incident ray. You employ the body, because it diverts rays of all colours by the same angle. A special design of spectral apparatus is discussed below. You can then link the collimator tube and the observation telescope rigidly and transit from one colour to another by merely rotating the prism.
Lummer and Brodhun 1889 have
employed the fact that a perfectly transparent, but totally
reflecting bounding surface can replace a wall which is totally opaque, in order to replace the photometer
screen of Bunsen by a more perfect one. Bunsen's screen
is imperfect, because
1. the opaque (not greased) part of the paper - it is to reflect diffusely the incident light, but not let any pass through - is not sufficiently opaque (both its sides are therefore lit by the incident light as well as somewhat by the passing light) and
2. the transparent (greased) part of the paper - it should not to reflect any of the incident light, that is, let all of it pass - is not transparent enough; it reflect a little, whence you see both sides in a mixture of both illuminations. Lummer and Brodhun have employed instead of the paper screen a combination of totally reflecting glass plates.
In Fig. 665, A and B are two right-angled glass prisms, halves of a glass cube. At qh, they are joined by putty with the same refraction index as their own. Otherwise they are separated by air. ll and ll are are two diffuse radiating planes. The light from ll is reflected totally at the air bounded part of the hypotenuse of B towards Y; in contrast, at the location qh, covered with putty, nothing is reflected and all incident light passes through. The light from ll , which meets qh, passes completely towards Y, but at the air covered parts of the hypotenuse of A it is totally passed on to X. Hence an eye at Y sees qh only in the light, which comes from ll, the surrounding part only in the light from ll.
The combination of prisms in Fig. 665 only explains the basic idea of the photometer screen. Lummer and Brodhun have built the photometer head in the forms A and B in Fig. 666, which shows also the layout of the entire photometer. The line mn represents the photometer bench, m and n are the sources of light to be compared. In between them, perpendicularly to the bench, is an opaque as white as possible plate ik; one side of it receives only light from m, the other only from n. The mirrors f and e throw the light, which is diffusely reflected from l and ll to A and B. The prism A has here a spherical surface and a circular area rs groiund on to it. It is pressed firmly with this circular area against B (without putty!).
If you view through the telescope W the area arsb, you see the image which you know from the Bunsen-photometer (Fig. 667): A circular area a, surrounded by the circular ring b, which is the more differently lit, the more the lighting of the two sides of the plate ik differs. At perfect equality of this illumination, you see a lit equally brightly as b, that is, a uniformly lit circular area. In order to illuminate the circular area and ring equally brightly, you shift the photometer screen along the bench (Fig. 623). It is 2.5 to 3.5 times as sensitive as Bunsen's photometer. The mean error of an adjustment remains below 1/2 % during measurement of a 50 candle lamp, that is, below 0.25 Candle.
Importance of the refractive index
Determination of the refractive index is one of the most important tasks of Optics. Microscope, telescope and photographic objectives owe their perfection to that of the art of melting glass, for which a knowledge of the index is indispensable. The refractive index of fluids has for Chemistry the same significance as other material constants; it frequently allows one to draw conclusions regarding the chemical structure of a body (molecular refraction). The purity of many fluids in use can be tested by means of their refractive index; for example, the purity of fats and volatile oils; milk is tested thus for its fat content, natural butter is distinguished from artificial butter.
There exist many methods and tools for the determination of the refractive index. We will describe two due to Ernest Karl Abbe 1840-1905: They form the basis for all later ones, also for those of Pulfrich 1858-1927, which are probably most commonly used. The spectrometer of Abbe serves the measurement with solid, transparent substances: It employs the principle of rays returning into themselves of Littrov 1781-1840. For the measurement with fluids, Abbe's refractometer is used: It employs total reflection.
Method of rays which return into themselves
Fig. 668 explains the determination of the refractive index with Abbe's spectrometer. ABA'B'EF is the prism of Fig. 633 (the substance which has been given for the purpose of the measurement this form), bca is a vertical section through its refracting edge EF, a principal section placed into the plane of the drawing; FS is a ray of light and takes the place of a bundle of parallel rays of light. We make the angle of incidence i, at which FS meets cb, so large that the ray after refraction meets ca perpendicularly. The corresponding angle of refraction is r. Its two legs (ray and normal) are then perpendicular to the two legs ca and cb of the angle d. Hence r = d. Since then sin i/sin r = n (the refractive index) and r = d, it follows that sin i/sin d = n, whence you measure the angles i and d for the determination of n. We have already described the measurement of the refracting angle of the prism. The return into itself of the ray coming out of the telescope can also be employed for the measurement of i (Figs. 659/660 above); the ray FST returns into itself if it meets the plane ac perpendicularly. Hence, in order to measure i, you rotate the prism from the position, at which the signal of light from cb returns into itself, until you note that the light from the plane ca returns into itself and read off the angle on the fixed graduated circle (Fig. 635).
Method of minimum deflection of a refracted ray
A method proposed by Fraunhofer is important: A bundle of parallel (mono-chromatic) rays enters the space (Fig. 669, seen vertically from above), horizontally and meets the prism with vertical, refracting edge. The bundle passes through the prism and - you should imagine that it is a pointer with end point F - and draws on the opposite wall a bright spot, which, as a result of refraction, does not lie on the extension of the incident bundle at X, but is deflected towards the base (to F1) by the angle s. If you turn the prism about the vertical axis through a, the spot F moves on the wall. If thereby the angle of incidence increases, as in Fig. 669 during the transition to Position 2 of the prism, the spot F moves towards the refracting edge to F2. Hence the direction of the emitting ray approaches that of the incident ray, that is, the deflection by the prism becomes smaller. However, starting from a certain certain position of the prism, the spot returns, that is, the deflection becomes again larger. At this position of return of the prism, the deflection is smallest (minimum position). The ray forms here (we omit the proof) inside the prism equal angles with the sides of the prism (V and W in Fig. 670) and the deflection angle d is related to the refracting angle L of the prism and the refractive index:
n = sin½(d + L)/sin½l.
The method is realized by means of a spectrometer (Meyerstein 1808-1882) with two telescopes (Fig. 671) for the incident and exiting rays. When in Abbe's method the ray returns to itself from the prism (Fig. 672), its inclination to the refracting plane is exactly the same as when it passes at minimum deflection through a prism with twice as large a refracting angle; hence also in Abbe's method, the measurement occurs in the minimum position of the prism (at minimum deflection, Fig. 673). - Fraunhofer's method is suitable for laboratories, Abbe's for workshops, both are equally accurate.
Method of total reflection. Abbe's refractometer
The refractive index of a fluid is normally determined by the method of total reflection. Abbe's refractometer involves for this purpose two right-angled glass prims A and B with known index n; they form together a right-angled parallelepiped (Fig. 674). You place in between the prisms a drop of the fluid to be tested. The index n of the glass must be larger than that of the fluid, whence you cut the prisms from glass with index n = 1.75, that is, larger than that of most fluids. If you now place the pair of prisms with the fluid between them in the track of a bundle of light the light does not always pass, indeed not when the angle of incidence, at which the boundary glass/fluid faces the light, reaches or exceeds the limiting angle of total reflection. Then the light is totally reflected as it transits from the more strongly refracting glass into the more weakly refracting fluid. The bounding angle g, at which this occurs, is indicated clearly by the refractometer (Fig. 676). This angle and the known index n of the prisms yields the index of the fluid: n = n sin g, for n is the index air-glass, n that of air-fluid, whence the index fluid-glass is n/n and for the bounding angle g of total reflection during the transition of light from the glass into the more weakly refracting fluid it is sin g = n/n. The scale of the refractometer, along which you read the bounding angle n, already yields the magnitude of n sin g (since n is constant), that is, you obtain directly the refractive index of the fluid.
Fig. 674 explains the basic idea of the refractometer. A and B are two right-angled glass prisms, C is the fluid to be tested, F is the lighted point from which rays reach the prisms, indeed parallel rays (with the aid of the fixed lens J). The rays pass through the (rotatable) pair of prisms AB as long as the angle of incidence, at which they enter from the (more strongly refracting) prism A the (more weakly) refracting fluid C, is smaller than the bounding value - however, they do not pass as soon as AB is rotated so far that this angle is reached. If the rays at all exit, they are parallel and have passed through the plane parallel body AB, they are parallel to each other. An eye at O receives the light from F and sees it like a far distant object. (Light reaches the eye only along the path with arrows.) However, apparently you only see the source of light as long as the light enters at an angle to the bounding plane C which is smaller than the bounding angle of total reflection; if you turn the pair of prisms in the direction of a growing angle of incidence, the wall becomes opaque as soon as the position corresponding to the bounding angle is reached - it becomes dark. The instrument measures the angle when this happens.
In the routinely employed refractometer (Fig. 675), the lay-out is as follows: 1. The eye and the source of light have been interchanged. The light passes first through the prisms and then through the lens. With reference to Fig. 674, this means that the light comes from the right hand side and F is the eye. This inversion of the sequence does not change the result (according to the principle of invertibility of the path of the ray). The eye sees in the tube the source of light O extinguished when during rotation of the prisms the angle of incidence reaches the limiting angle. 2. The rays of light meet the layer C in all possible directions. But only those can pass, the angle of incidence of which is smaller than the limiting angle of total reflection. If you have turned the prisms to the position at which the ray, which falls parallel to the axis of the telescope on B, meets C at the bounding angle, then one half of the field of vision - we do not explain why - receives no rays whatsoever and is dark, the other is bright, whence it looks like shown in Fig. 676. The operation of the refractometer is now obvious: You place a drop of the fluid to be tested between the two prisms, apply with the mirror (homogeneous) light, look into the tube, rotate the prisms with the knob until the field of vision becomes like shown in Fig. 676 and obtain through the magnifying glass the index. (If the light is not homogeneous, the subdividing line has colours and is faded. There is a special arrangement for removing the colour).
You can display directly, when white light passes through a prism, that the rays corresponding to different colours are refracted differently and that thereby form colour dispersion. However, colour dispersion is only measurable by the refraction indices for the individual colours and the relations between them. The glass industry has special interest in fining out the magnitude of dispersion: The suitability of a glass for microscopes and telescopes (especially avoidance of coloured edges around images) is largely determined by its path of dispersion.
Following Abbe, you specify the properties of kinds of glass by the refraction indices for the lines A'.b C, D, F, G', more recently also for the helium line d and the mercury lines e, g, h. The path of dispersion is characterized by the refraction differences for the sections A' - C, C - e, e - F, F - g, g - h as well as the ratio of these partial dispersions to the average dispersion between the lines C and F.
You try to express the relationship between a wave length l and the corresponding refraction ratio nl by means of a dispersion formula. You confirm the correctness of the formula by checking the computed nl against the value obtained from a spectrometer. The oldest formula, due to Cauchy 1836, for normally (in contrast to anomalously) dispersing, isotropic transparent substances is nl = a + b/l2+ c/l4 + ···, where a, b, c ··· are constants which characterize the substance and must first be determined. For most substances, nl = a + b/l2 is sufficient and agrees with measurements. Hence, if you measure with a given substance for two definite wave lengths (colours), for example lC and lF, the refraction ratios and compute from the equations for nC and nF the constants a and b, you can compute for every given l the corresponding nl with sufficient accuracy and till agree sufficiently with a control measurement.
However, Cauchy's formula proves to be true only for wave lengths between 0.4 and 0.8 m (within the visible spectrum). If you compute from it the refraction numbers for longer waves (in ultra-red), the formula is less useful even near the visible spectrum and becomes less for longer wave lengths, for example, for water, glass, quartz, fluor spar, rock salt. Modern dispersion theory, which employs electro-magnetic theory of light yields results which are much closer to experimental results. It explains refraction by interaction between the ether and the molecules of a substance, respectively the electrons linked to the molecules, that is out of the influence of the molecules on the vibrations of the ether. The simplest such dispersion formula is n² = b² + M1/(l² - lv²) + M2/(l² - lr²) (Eduard Ketteler 1836-1900 1893 and Helmholtz 1893), where M1 and M2 are depend on the nature of the dispersing substance and interlinked closely to its dielectric constant (cf. below); lv and lr are are two wave lengths, characteristic for the substance in the ranges ultra-violet and ultra-red. The substance absorbs them, because they excite the electrons in the molecules (resonance); they can be determined exactly experimentally. For many substances such as water, rock salt, sylvine, fluor spar (Ernest F. Nichols 1869-1924, Heinrich Rubens 1865-1922), the formula represents dispersion exactly. For example, for quartz, it is insufficient and demands a further extension.
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