Geometrical optics

L2 Brightness

Transparency and opacity

Light which spreads without encountering obstacles (the retina being one of them) is imperceptible. When it encounters an obstacle, say a wall, it becomes perceptible by phenomena which differ with the kind of wall, which can more or less let the light pass . We call a wall transparent which lets light pass, and in doing so we relate to our eyes. The more transparent is a wall, the harder it becomes to see it. Frequently, it is difficult to discern whether an opening through which one looks has glass in it or not! If the glass is especially clear and without colour and no reflection hits the eye, it cannot answer this question. Visibility demands a certain degree of opacity, whence in a camera viewer clear glass in place of opaque glass would be senseless.

If a wall does not let light pass, is is said to be opaque. There are many transitions from transparency to opaqueness. We call a substance translucent (diaphanous), if we can neither call it transparent nor opaque. The degree of transparency depends on the material of the wall and its thickness. If you make it sufficiently thin, it lets light pass even when the material is paper thin and opaque like silver or gold. There exist perfectly transparent metal foils out of nickel, gold, platinum, silver, iron, some of them 1·10-6 cm, that is, about 25 atomic layers thick. In contrast, materials which we ordinarily know to be transparent become opaque when they are sufficiently thick - transparency as well as opacity are relative.

The theory of waves assumes that opaque materials absorb the energy of entering ether waves or rather convert it into another form of energy - not light but, may be, heat; in contrast, transparent materials let the waves of light pass. Hence the higher or lower degree of transparency of a substance is explained by the degree to which it absorbs waves of light which penetrate it.

Reflection of light

A wall can pass or absorb light. However, there exists a third possibility. Most of the objects we see are not themselves source of light, but, in order to become visible, they demand the co-operation of a source of light. Objects which we see in a room lighted by a lamp become invisible to us when the light is switched off. We see them as soon as and as long as the light is turned on. Hence the objects, although not sources of light on their own, send out light as soon as and as long as light meets them. We say that they are illuminated and that they reflect light; depending on the strength with which they act on our eyes, we say that they are more or less brightly lit.

The phenomenon, by which light tells us that it has been reflected, can differ very much; for example, if the wall is a mirror or a piece of white cloth, covering the mirror. A mirror reflects the light, which comes to it from a certain direction, in a certain direction in a simple manner according to a definite law; the cloth does so irregularly in all directions. The light coming from a mirror changes for us as we move in front of it and look into it in different directions. In contrast, the white cloth does not change from wherever our eyes look at it. However, in the case of the mirror, we do not see the reflecting surface, only images. We see an area only when it reflects the light in such a manner like the white cloth; materials which are neither completely transparent or completely reflecting behave like this. We could equally well say: In order to be able to be illuminated, that is, to become visible, a wall must be to same extent opaque and dull. - The light sensation, caused in our eye by a body which is opaque like the white cloth and returns the light diffusely, is merely brightness - altogether the simplest sensation of light. Other additional impressions which the eye gives us like form, size, disappear in the process.


We understand here by brightness the sensation of light familiar to everybody. However, you should not confuse physiological brightness, for example of an area, with physical brightness, that is, the state of an area on which light falls from a source. A source of light has a certain intensity of light, emits a flow of light in all directions, creates on the area an intensity of illumination (illumination) - all these are physical states which can be measured in agreed units - and this area appears to us to have a certain grade of brightness.

The intensity of light (J) is considered to be a fundamental quantity, because its unit can be realized by a source of light (Hefner-candle). A point-source of light, as centre of a sphere, emits flow of light (F) in all directions. It spreads in spherical waves into space and generates on a spherical surface a uniformly distributed intensity of illumination (E). This intensity is inversely proportional to the square of the radius of the sphere (that is, inversely proportional to the square of the distance of the illuminated surface from the point-source), because the spheres with radii r1 ···rn have r12 ···rn2 times as much area as the sphere with radius 1. If the light generates per cm² the light intensity E1, it generates on the spherical surfaces with radii r1 ···rn the intensity E1/r12 ··· E1/rn2 (where we define the intensity of illumination in such a way that its numerical value is equal to that of the intensity of light, when the perpendicular distance between the source of light and the illuminated area is 1 cm.)

In reality, rays of light do not always meet the illuminated area like the radii of a spherical surface. If the illuminated area (F') is at an angle to the flow of light (Fig. 616), the area ab2 is met by a smaller bundle of rays than if it is perpendicular to it. If F deviates by the angle j from the vertical to the bundle, ab2 receives only as many rays as it would receive in the perpendicular position from ad = ab2 ·cosj . The angle is equal to the angle of incidence a, which the rays form with the perpendicular to ab2. Hence follows Lambert's fundamental photometric law: The intensity of illumination E is proportional to the cosine of the angle of incidence of the rays and inversely proportional to the square of the distance from the light's source, that is, for a point source, E=(Jcosj/r2).

Equally important is the angle between the rays and the normal to the illuminated area, when its element is the source of light and emits rays (a red hot sheet of platinum). You then talk of the luminous density (e). If the area element s radiates at the angle j (to its normal) the intensity of light J, you understant by area brightness in this direction e = J/s cosj (Fig. 617). The denominator is the projection of the area element s on to the plane, perpendicular to the direction of the rays. Correspondingly, the intensity of light in this direction is J = es cosj. - You have perpendicular to s the luminous density e = J/s.

You can readily convince yourself of the importance of the angle j for illumination and brightness. Without being conscious of it, you take it into consideration when you read in the light of a distant and fixed lamp and incline the book so that it is illuminated as well as possible.

You understand by brightness of the small, uniformly illuminated area s (Fig. 618) the ratio of the stream of light F, which is radiated by s to the pupil of your eye and further transmitted to the retina, to the size s' of this image, that is, the illumination of the retina at the location of the image is h = F/s'. The brightness h of s is computed independently of the distance to the eye and proportional to the luminous density e in the focussing direction. Indeed, a uniformly luminous body, for example a red hot platinum sheet, appears to be uniformly bright whatever is its location with respect to your view and whatever is its distance from your eye; if it is curved, you cannot tell it apart from a flat sheet. Hence a red hot metal cylinder and bar give the same impression and so do a luminous sphere and disc. (A similar result applies to heat radiation and is important for the law of heat exchange.)

The unit of light intensity (J) is the candle of H. von Hefner-Alteneck (HK): It is the horizontal intensity of the light of the Hefner-lamp, which employs amyl acetate with the 40 mm height of the flame. The unit of the stream of light (F) is that emitted by a 1 HK point-source in all directions in the space angle 1. This unit is called 1 Lumen (Lm). [The space angle 1 is the angle at the vertex of a cone which is cut from it by a sphere with radius 1 m so that the corresponding piece of the spherical surface is 1 m².] Hence a source of light which emits in all directions with the intensity J radiates the total stream of light F = 4pL; for J = 20 HK, F = 251.3 Lm. The unit of illumination has an area which is illuminated vertically by 1 HK at a distance of 1 m (according to the formula E = Jcosj/r²). The unit is called Lux (Lx). A source of light J = 25 HK generates on a 2 m distant area for j = 0 the illumination E = 25/4 = 6.25 Lx, for j = 60º, the illumination E = 3.125 Lx (since cos 60º = 0.5). The unit of surface brightness (e) has, according to the formula e = J/scosj, an area which radiates per 1 cm² vertically (j = 0) the light intensity 1 HK. Surface brightness in candles per cm² is for a kerosene lamp about 1, for an incandescent gas lamp 5 - 6, for a carbon filament lamp (4 Watt per HK) 48, a metal filament lamp (1.1 Watt per HK, a zig-zag shaped wire in vacuum) about 150, for the arc lamp crater about 18,000.

Measurement of light intensity (Photometry)

While the eye cannot assess by how much brighter one area is illuminated than another, it can decide whether two adjacent areas are equally brightly lit. Its accuracy depends on the magnitude of the area brightness and is largest if the brightness of the areas to be compared is about the same as that of diffuse daylight. It then recognizes the areas, if they are illuminated by light of equal colour, as differently bright, when the difference is only 2/3 % of the brightness. - We draw from the equality of the brightness of two illuminated areas a conclusion regarding the illumination intensities, and compute from the distances, which the sources of light must have from those areas in order to cause the observed brightness, the intensity E = (J cosj)/r² of the sources of light. Thanks to the competition between the various technical kinds of illumination, photometry and its instruments (photometers) have reached as state of perfection.

Only the eye can decide immediately whether two sources of light are equally bright. However, waves of light act also thermally, electrically and chemically. All these phenomena are employed in special cases of objective photometry. The photo-effect of the alkali cell is used in the photometry of incandescent lamps, since brightness is strictly proportional to the electric current; it is claimed that optical and photo-electric measurements agree to within 1 %. - We are here only concerned with subjective photometry.

In essence, the aim of optical photometry is to exhibit to the eye two adjacent planes, one of whichonly receives light from the one, the second from the other light. The photometer of Bouguer is very simple (Fig. 619). You must ensure that both areas are illuminated at the same angle, for the intensity of illumination depends essentially on the angle at which the rays arrive. Fig. 620 shows how the photometer of W. Ritchie meets this condition. It is simplest of all to let the rays fall perpendicularly on the two areas by placing the two sources of light I1and I2 on both sides of an opaque white plate S in such a manner that a straight line joining the two sources of light is perpendicular to the centre of the plate. If you then change the distances r1 and r2 of the sources of light from the plate, it is readily achieved that both sides of the plate appear to be equally lit. This equality leads to the conclusion that the intensities of the lights are equally strong, that is,

I1/r1² = I2/r2², whence I1/I2 = r1²/r2²,

that is, the intensities of the two sources of light are related to each other like the squares of their distances from the illuminated plate.

You will object: "The illumination is only equal to I/r² when I is a point source of light and it illuminates a spherical surface with radius r; however, in reality, the source of light is a shining plane and the area illuminated in the photometer is a plane." This defect is already without consequence, when the distance of the sources of light is about 10 times as large as their largest dimension and when the straight line through their centres passes approximately through the centre of the illuminated plate.

You must make sure that the eye sees the two illuminated planes at the same angle as in Ritchie's photometer, because the brightness of an area depends on the angel at which you see it. In order to see both side by side, you place the plate into the angle of an angled mirror. - You manipulate the photometer as follows: Place on a bench the sources of light on both sides of the plane to be illuminated - the photometer screen S - so that the straight line. linking the centres of both sources of light, is perpendicular to the centre of the screen, adjust the distances of the sources of light from the screen so that both sides are equally brightly lit, and measure the distances. Denoting the distances by a1 and a2, the strengths of the sources of light by I1 and I2, then I1/I2 = a1²/a2². For example, if a1 = 100 cm, a2 = 141.4 cm, then a1²/a2² = (1/1.414)² = 1/2, whence I2 = 2I2.

Bunsen photometer

The principle of the photometer of Bunsen 1848 is the foundation of the most frequently employed equipment. The (antiquated) photometer screen, on the sides of which you place the sources of light, is a sheet of white paper with a circular grease spot at its centre. This greased spot allows more light to pass through than the rest of the paper; because it behaves differently to the surrounding paper, it appears to be very characteristic depending on the illumination. If you look at a grease spot on a piece of paper, while your eye and the source of light - say the window - are on the same side of the sheet of paper, that is, if you look at it in incident light, the spot is known to look darker than its environment. However, if you look at it while the eye and the window are on opposite sides (the sheet is now between your eye and the window), that is, you look at the sheet in light which passes through the paper, the greased spot appears to be brighter than its environment. Why?

Only one side of the paper receives light (we can disregard the diffusely transmitted light from the walls of the room; besides it meets both sides of the paper). The section of the paper, which has not been greased, effectively lets no light pass, it reflects the light diffusely in all directions on the side from where it came, but the greased spot allows light to pass to the other side and reflects only very little. If your eye is on the side of the source of light, it receives therefore very much light from the surroundings of the greased spot, but comparatively little from the spot itself, whence the greased spot appears dark on a light background. If your eye is on the other side, it receives comparatively much light from the greased spot, but little from the almost opaque paper; you see the spot lighted up against a dark background.

If both sides of the paper are equally brightly illuminated, the greased spot cannot stand out of its environment, it becomes invisible. (Actually, it does not become completely invisible; it retains a minimum of visibility, but this minimum appears to be equally strong on both sides of the paper.) In order to see both sides next to each other, you place behind the screen an angled mirror, so that the screen lies symmetrically with respect to the two mirrors. You change the distances of the sources of light to be compared with each other until the greased spot has on both sides minimal visibility. The distances yield then the ratio of the intensities of the sources of light.

Since 1889, use of the photometer screen of Lummer and the experimental lay-out of Brodhun has yielded better results; it employs total reflection.

Ball photometer

Only a point-source of light emits in every direction the same intensity, whence it only demands a single measurement on the photometer bench. Every other source of light demands measurements in every direction, if you want to know their distribution of light into space. However, that is technically impossible. Hence you evaluate every technical source of light according to the stream of light F = 4pJ0 (where J0 is the mean spatial light intensity). You find it by means of a single measurement in the ball photometer of Ulbricht. The lamp L hangs in a dull white painted hollow sphere K (Fig. 622). A dull white screen S protects the observation window o - an opening in K closed by a milk glass disk - against direct illumination by L. If the reflection of the painted surface follows Lambert's law, the illumination at o, that is, the indirect illumination from the wall of the sphere through L, is proportional to the total light from L. Let F denote the required total light emitted by L, E the illumination in o, then F=KE, where K is the constant of the sphere, which you obtain by replacing L by a normal lamp. You can measure the illumination at o by determining the intensity of the light of the opalescent glass plate with a bench photometer.

Unit of light intensity. Normal candle

You can measure with a photometer whether two given sources of light are equally bright or how much brighter is one of them. However, you then still do not know how bright it is. There is a need for a unit of light, on which measurements of light must be based just like measurements of distance are based on the centimetre. This unit of light had to be fixed arbitrarily*. In Germany, it is the Hefner-candle (normal candle), the light intensity of the amyl acetate lamp (A in Fig. 623) introduced by H. von Hefner-Alteneck. The dimensions of the lamp and the wick are strictly prescribed; the lamp is filled with chemically pure amyl acetate; if it burns (following indispensable safety measures, which concern the state of rest and purity of the air) with a 40 mm high flame, then the intensity of the light is 1 candle (HK). An electric lamp has 16 HK means: It is 16 times as bright as the Hefener candle.

* The unit of light intensity in the USA, England and France is the international candle (= 0.9 HK).

If you place a Hefner lamp A at the one end of the photometric bench (Fig. 623), the light source L, the light intensity of which is to be measured, at the other end, displace the photometer head P until its screen looks equally bright on both sides, you determine the number of HK of L. For technical purposes, you employ instead of the Hefner candle stronger sources of light, the intensity of which has been previously determined in terms of Hefner candles, especially electric incandescent lamps.

For the illumination industry, it is important to find out the strength of illuminations at a given location. As a measure, you employ the illumination of a candle at a distance of 1 m. Such a unit of illumination (1 Lux) is required for regulations relating to the light at a working place (reading, drawing, sewing). 60 Lux are sufficient for reading as fast as in daylight, a common working site demands at least 12 Lux (Hermann Cohn). - The intensity of illumination is measured with specially designed photometers (Weber, Bechstein).

Brightness of stars (apparent)

Stars are ordered according to their brightness in classes of magnitude [photometric magnitude (m)]; it has been agreed internationally** that such an entire photometric class separates two stars from each other, the brightnesses of which are related as 1 : 2.5 (strictly speaking, 1 : 2.512 for more covenient employment of logarithms: log 2.512 = 0.41!).According to this definition, J1 = 2.512xJ2, J2 = 2.512xJ3, that is, for example, J1 = 2.5124xJ5. If two stars have the brightnesses Jp and Jq and the class magnitudes mp and mq, then

Jp/Jq = 2.512q-p, whence q - p = (logJp - logJq)/0.4.

For example, for p - q = 5 (class magnitudes), 5x0.4=logJp/Jq or Jp/Jq = 100.

** Within a large brightness interval, the smallest perceptible differences dE (of subjective sensitivity of light E) correspond to almost constant fractions dH/H (of the objective brightness H). This kind, which is as the eye measures brightnesses, has influenced the ordering of the magnitudes of stars (Fechner , the originator of the fundamental psycho-physical law dE = k dH/H, which has also proved useful in other areas of psychic sensing). Without measuring the objective quantity of light, the magnitude classes of stars have been determined according to the impressions which they make on the eye; you calls the brightest star to be of magnitude 1, the next bright ones of magnitude 2, and those, which are just perceptible to the eye, of magnitude 6.

Hence a star of first magnitude (p) is 100 times as bright as a star of magnitude 6 (q). (The zero of the visual scale of the Harvard Observatory has been defined by the Polar Star being +2.12m.) The definition above does not only yield fractions of magnitude classes, but also negative numbers. The brightest stars - Sirius and Canopus - have the magnitude -1.6 and - 0.9, the Sun - 26.5. The weakest stars, visible to the eye, are in the class 6, those visible to the best telescopes in the class 17, the photographically detectable ones in the class 20.

Following Johann Karl Friedrich Zöllner 1834-1882, you measure the brightness of stars by the impression on the eye, by adjusting to equal brightness with an artificial star, the brightness of which you can alter; following Wilsing, you extinguish by means of a measurably changeable layer of a light absorbing medium or assess it from the impression on a photographic plate or a light-electric sensitive cell. However, the results of these impressions are not identical, because each of them is created by a different section of the spectrum. The difference of thus obtained brightnesses of the same star - especially the difference between the photographically and visually determined brightness - is called the colour index. - Two stars may appear to the eye to be equally bright, but lie apart photographically by two classes of magnitude, because a red star acts chemically more weakly. Hence the colour index characterizes the spectral distribution of intensity and hence the temperature; it is closely related to the spectral type. - In order to obtain agreement between the visual and photographic scales, it has been agreed internationally that, for the stars of magnitude 5.5 - 6.5 from the spectral type Ao (Sirius), the photographic and visual magnitudes are to be considered to be equal.

You distinguish between apparent and absolute brightness - the apparent one is sensed by the observer, the absolute one which he would sense at the distance one from the star - distance one corresponds to the parallax 0.1". For example, the Sun appears to be 6x1010 times as bright as Capella (26.9 magnitude classes brighter); if you set the parallax of Capella at 0.08", since the apparent brightness decreases quadratically with the distance, the calculation yields that the Sun at the same distance from us as Capella would appear to us to be in the magnitude class 5.4. - In general, the concept of absolute brightness is only applied to fixed stars which glow on their own. If M denotes the absolute magnitude, m the apparent one, p the parallax, then M = m + 5 + 5log p .

continue step back