## Illumination

### Introduction

In most optics texts, illumination is either not mentioned, or is given only cursory treatment. The probable reason for this is that illumination involves the psychophysics of the visual sense, which may seem out of place in "straight" physics. However, illumination is not only of considerable practical importance, but its definitions and methods are applicable to the transfer of any kind of radiant energy. The strange units of illumination may be regarded askance, but a basic unit, the candela, is a fundamental unit of the SI system. The theory of illumination involves only the cosine factor for projecting areas, and the inverse-square spreading from a point source. Its practical results are expressed as surface integrals, which can now be done numerically with computer aid. The theory, therefore, is quite simple and easily mastered. Names and definitions often create confusion, however, which this article will strive to overcome.

The measurement of the energy of radiation, an objective quantity that can be measured in W, is called radiometry. when the spectral sensitivity of the eye is taken into account, the measurement is called photometry, where light is measured in lumens. Photometry is semi-objective, intermediate between the physical stimulus of energy and the psychophysical response of brightness. Although we often say "eye", the visual sense is actually located in the brain; the eye is merely a sensor.

The results of illumination theory will be applied to two important theorems about the intensity of an image formed by an optical system. One is that the brightness of the image cannot exceed the brigtness of the extended source that is imaged, and the other is that the illumination in the image decreases as the fourth power of the cosine of the angle of the principal ray (the one through the centre of the entrance pupil).

### Lumens

We use the same word "light" for electromagnetic radiation of frequencies in the narrow band 4 x 1014 Hz to 8 x 1014 Hz, and also for the psychological sensation produced by it when it impinges on our eyes and excites our visual sense. The energy in physical light can be expressed in watt, which is precisely defined. Its value in producing sensation, the strength of which is called "apparent brightness," is less well defined because of the difficulty in the quantitative evaluation of sensation. Nevertheles, by averaging the responses of many observers, a curve of the relative efficiency of energy at different spectral wavelengths in producing sensation can be determined. The result is called the "Standard Observer," whose spectral sensitivity is plotted at the left. The peak of this curve is at 555 nm, taken as unity, and is down to 0.0004 at 400 nm and 735 nm. The commonly used visual range of 380-760 nm includes a lot of worthless "tail" region. A better statement would be 500-630 nm, showing how narrow the eye's spectral response really is. A quantity called luminous flux, F, is defined that is analogous to energy, but reflects the effectivness of the radiation at producing visual sensation. This unit is the lumen, and at the peak of the photopic (light-adapted) eye's sensitivity, 680 lm = 1 W (some references give 683; the difference is inconsequential). Now we can convert any spectral distribution of energy into lumens with precision, and work with lumens as we would work with energy. It is only necessary to multiply the energy in watts in each small wavelength interval by the visual efficiency, and sum the results, multiplying by 680 or 683 to get the lumens.

It must be carefully appreciated that lumens do not measure brightness, which is like loudness in acoustics. Establishing a scale of brightness is a completely different matter, and one that belongs exclusively to psychophysics. All we know is that equal amounts of luminous flux produce equal brightness, and more flux means more brightness, but no more than that. In fact, brightness is about proportional to the logarithm of the luminous flux (Fechner's Law). Brightness could be defined by the relation B = k log(F/F0), where we would have to choose a constant k and a reference luminous flux F0. Doubling the luminous flux does not double the apparent brightness. The term "brightness" was once used for certain photometric quantities, but now has been replaced by "luminance" to avoid confusion with psychophysical brightness. The eye can, however, detect equality of brightness quite reliably, and this property is a valuable one. It is impossible to say when one surface is twice as bright as another, so establishing a quantitative scale of brightness is difficult.

We may proceed as we have done for lumens with any similar weighted energy distribution, or with energy itself, in what follows. Some of the names used, however, are peculiar to illumination and lumens, and should not be used with energy or other radiant quantities. The Latin word lumen, luminis (n.) is one of two words meaning "light." The other is lux, lucis (f.). Lumen was often thought of as light coming from the eye, or a lamp, while lux was light coming into the eye, or from the sun or moon. Both these words are used in photometry to name concepts and units. Light measured in lumens may be monochromatic, but the concept is really intended for use with broad-spectrum light, often perceived as white.

### Intensity

Now let us consider a source of luminous flux, and a specially simple one that has no spatial extension, but emits luminous flux along radial lines. This point source need not be equally strong in all directions, and can be as anisotropic as desired. Any finite amount of radiation must be emitted in a finite cone surrounding the direction considered, that can be made as small as desired. This cone has its vertex at the source, and its base of area dA at a distance r from the source, the normal to dA making an angle of φ with the radius. Then, this cone is measured by the quantity dΩ = dA cos φ/r2 called a differential solid angle, measured in steradians. The definition is illlustrated at the right. It is positive or negative as the normal to dA points outwards or inwards. It is clear that the total solid angle surrounding a point is 4π.

The luminous intensity I of a point source is the ratio dF/dΩ, and is in general a function of direction. It is measured in candela, cd. If 1 lm is emitted per steradian, the intensity is 1 cd. An isotropic point source of intensity I cd, then, emits 4πI lm. The candela is not far from the actual luminous intensity of a normal candle flame, and was once defined in terms of standard lamps burning pentane, amyl acetate, or colza oil. These days it is the intensity of an area of 1/60 cm2 of a black body at 2042K (freezing platinum). A 60W gas-filled tungsten incandescent lamp provides about 870 lm when new. This corresponds to 14.5 lm/W referred to the electrical input power to the lamp. If the lamp radiated uniformly, its luminous intensity would be 69 cd. The specification of lamps by candlepower was once common, but it is easier just to give the electrical input if you want to make substandard lamps. Actually, both should be given to estimate the balance between life and efficiency. A very efficient lamp will burn hot and expire sooner from evaporation of the tungsten. Long-life lamps are easily made by simply reducing the lumens per watt. A 400W high pressure sodium arc gives 50,000 lm, or 125 lm/W, about twice the efficiency of a fluorescent lamp. An Edison carbon-filament lamp gave about 3 lm/W. This efficiency should not be confused with the visual ratio of 680 lm/W, where the energy is already in the form of radiation. If all the energy input to a lamp were output at 555 nm, then its efficiency would be 680 lm/W, which we can regard as a kind of upper limit, never closely approached. The renaming of the time-honored luminous efficiency to "luminous efficacy" is yet another example of worthless pedanticism.

### Illumination

The luminous flux falling on the area dA from a source of intensity I is given by dF = IdA cos φ/r2, as shown in the diagram at the left. This follows directly from the definition of I as luminous flux per unit solid angle and the definition of solid angle. If the source is an extended one, then this must be integrated over the source area. The luminous flux per unit area falling on a surface is called the illumination E of the surface, and is measured in lm/m2. A lm/m2 is called a lux, and a lm/cm2 is called a phot. Clearly, 10000 lx = 1 phot, for what it is worth. For a point source, E = dF/dA = I cos φ/r2.

All that is involved here is an intensity in cd and a distance. We get different units if we take the metre, centimetre and foot as the distance units. So, in addition to the lux and the phot, we have the ft-cd, foot candle, which is lm/ft2. It is easy to convert between these units, but it would be less confusing to use the full dimensions rather than the given names. I get 1 ft-cd = 10.76 lux. See if you agree. 30 ft-cd or 300 lux is considered adequate for normal work. A ft-cd (fc) is 0.929 milliphots, by the way.

The term illuminance has been proposed to replace illumination, apparently to show that the word has a technical meaning that should be distinguished from the general term illumination. This is yet another useless complication, with much less reason than the substitution of luminance for brightness. It might even create some confusion with luminance, which sounds similar. Nobody confuses illumination with lumination, and if one does, it is harmless.

### Luminance

Most of the confusion in illumination calculations now comes when we consider the illuminated surface as a new source of luminous flux. Illuminated surfaces differ greatly in their response to incident light. A specularly reflecting surface, such as that of a metal, reflects the light according to the laws of reflection. A surface may be perfectly absorbing, or black, and in this case it just soaks up the luminous flux and does not return any. Most surfaces are somewhere in between. The science of illumination mainly concerns itself with the ideal case of a diffusing surface as defined by Lambert. Such a lambertian surface does not lose any incident radiant flux, but re-emits it in all the available solid angle, which here is 2π radians, on the illuminated side of the surface. Moreover, it emits it so that the surface appears equally bright from any direction. That is, equal projected areas radiate equal amounts of luminous flux. Though this is an ideal, many real surfaces approach it.

We consider, then, an infinitesimal area dS of a lambertian surface emitting luminous flux at an angle θ with its normal, into solid angle dΩ. Then, d2F = BdS cos θ dΩ, where d2F is written to indicate that it contains two differentials, dS and dΩ. The factor B is a constant for a lambertian surface (it may vary with θ for a more general surface). The illumination of an element of surface dA by an element of bright surface dS is shown in the diagram. The expression involves only cosine factors and the inverse square spreading, so it should be easy to understand. The letter B suggests brightness, which was its original name, but possible confusion with the psychophysical brightness has led to its renaming as luminance. Since dF/dω is measured in cd, B must be measured in cd/m2, cd/cm2 or cd/ft2. The cd/m2 has been named the nit, and the cd/cm2 the stilb by the enthusiasts for unit names, not altogether felicitously. The nit is in disgrace, but the stilb appears to be officially sanctioned. The name comes from the Greek stilbw, "I shine." Nit comes from Latin niteo, also meaning "I shine." Nit is also the larva of the head louse.

If we integrate over dS (presuming dΩ remains unchanged), we find dF = I dΩ, where I = ∫B cos θ dS. If θ is also about constant, then I = B (S cos θ) = B x projected area, which makes clear Lambert's definition of his ideal diffuse reflecting surface. If we are looking normal to a disc of radius a and luminance B, then its intensity is πa2B cd. The illumination at a distance r will then be E = π(a/r)2B. The angular subtense of the diameter of the disc is 2a/r = δ. Therefore, E = (π/4)δ2B. When finding the illumination due to an extended source, dI = B cos θ dS.

The luminance of the sun is about 1.6 x 109 cd/m2, and its angular subtense is δ = 0.5° = 8.73 x 10-3 rad. Therefore, E = 96,000 lux on a surface normal to the sun's rays, or 62,000 lux on the level ground when the sun's elevation is 50°. The moon's brightness is only about 2500 cd/m2, so it illuminates a surface normal to its rays with 0.15 lux. The remarkable adaptation of our eyes to the full range of natural illumination is much to be admired, and gives a good reason for logarithmic response. 120 lux is the geometric mean of solar and lunar illumination, and this is about the lower limit for comfortable vision. The luminance of a 400W high-pressure sodium lamp is 780 cd/cm2.

The total luminous flux E emitted per unit area from a lambertian surface of luminance B is easily calcuated. ∫(0,π/2) cos θ dΩ = 2π ∫ cos θ sin θ dθ = π, so E = πB. The construction of this integral is shown at the left. Note that the radius of the hemisphere is immaterial. An area of luminance 1 cd/m2 emits π lm/m2. This factor of pi should cause no confusion if its source is kept in mind. However, there are other units of luminance B that include it. A surface with a luminance of 1/π cd/m2 emits 1 lm/m2. This amount of luminance is called an apostilb, confusingly changing from centimetres to metres, so a square metre of lambertian diffuse radiator radiates a total amount of lumens equal to its luminance in apostilb. Similarly, 1/π cd/cm2 is a lambert, and 1/π cd/ft2 is a foot-lambert. In Greek, apo means "away from," so apostilb is "I shine out." We should have consistently used apostilbs for lamberts, and "exnits" for cd/πm2. All this Greek and Latin is interesting, but I prefer to use only lumens, candela and the distance unit so I can keep things straight. The factor of π applies only to an ideal lambertian radiator, of course. If you assume that a surface reradiates all the luminous flux that falls on it, then its luminance in apostilbs, lamberts or foot-lamberts is the same as its illumination in lux, phot or ft-cd.

Since the idea of lamberts may be confusing, perhaps another description would be welcome. Suppose you are looking at a small illuminated diffuse reflector of area dA from a certain angle, and receive a flux of dF lumens from it. The projected area normal to your line of vision is dScos θ. If you look at it from a different direction, the projected area may change, but the area will look equally bright, which means the same flux per unit projected area. We may also introduce the solid angle dΩ of your pupil to find the flux per unit solid angle as well, which will allow us to integrate the flux over any surface. Then, our observation is that dF/dΩdAcos θ equals a constant, say L, so that dF = Lcosθ dAdΩ. To find the total light emitted by dA, we integrate over dΩ=2π sin θdθ from θ = 0 to π/2. The result is dF = πLdA, so the constant L is L = (1/π)dF/dA = E/π, or the total flux emitted (which will be a fraction of the total illumination) divided by π. A lambert is a lumen/cm2 received and reemitted per unit solid angle dΩ, not the luminance B, which is also lumen/cm2, but directly emitted into dΩ.

If a diffuse surface receives E lumens/cm2, then E/π is its surface brightness in lamberts, and the light emitted at an angle θ into solid angle dΩ is (E/π)dAcos θ dΩ. The total light emitted from dA is then E.

### Brightness in Images

We have now defined the four main illumination quantities: F, I, E and B, and given the connections between them. It is good to remember that I = dF/dΩ, E = dF/dA and B = d2F/dAdΩ. We will now look at some important properties of the illuminance of images formed by optical systems. In optics texts, this is usually called "brightness," but we have explained above why this term has been generally replaced by "luminance." The argument can be made rigorous, but we shall be satisfied with a simple demonstration that emphasizes the principal facts.

As shown in the diagram, a lens L forms an image I of an object O. dA and dA' are elements of an extended source and image. The rim of the lens is the entrance and exit pupil of the system, defining the extent of the pencil of rays that passes through it. Since the magnification y'/y = -s'/s, dA' = (s'/s)2 dA. The solid angle subtended by the entrance pupil at the object is Ω = πh2/s2, while the solid angle subtended by the exit pupil at the image is Ω' = πh2/s'2. Therefore, Ω'/Ω = (s/s')2. If B is the luminance of the object, and B' the luminance of the image, then the luminous flux in the input is BΩdA, while the luminous flux in the output is B'Ω'dA'. If there are no losses between source and image, these quantities must be equal, or BΩdA = B'Ω'dA'. This means that B'/B = (Ω/Ω')(dA/dA') = 1, or B' = B. The image luminance is equal to the object luminance.

The reason for this is clear. If the image becomes smaller, so that the same energy is concentrated in a smaller area, the solid angle under which it is illuminated increases proportionately, so the product remains constant. If the image is viewed by the eye so that the entrance pupil of the eye is full, the luminous flux entering the eye will be constant, equal to the image brightness times the solid angle subtended by the eye pupil.

If the image is formed on a diffusing screen, the same total luminous flux will come from a smaller area, which will appear brighter to the eye. A small image of the sun may ignite tinder if its temperature is raised enough, but this does not mean that the actual image has a greater luminance than the surface of the sun, but only that the energy comes from a larger solid angle.

The illumination in an image (not the luminance!) falls off for off-axis image points. If Ω is the solid angle on the axis, say A/s'2, the solid angle off the axis at an angle θ will be Ω' = (A cos θ)/(s'/cos θ)2 = Ω cos3θ. Since the illumination now falls obliquely at an angle θ, there is a further factor of cos θ. The illumination BΩ' = BΩ cos4θ. Therefore, the off-axis illumination falls off as cos4θ, which can be rather rapid. At only 20°, the illumination is off by 22%.

### References

Handbook of Chemistry and Physics, 56th ed. (Cleveland: Chemical Rubber Publ. Co., 1975). pp E-204 to E-208 and E-247.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 2nd ed. (New York: McGraw-Hill, 1950). pp. 104-111.