Radiometry and photometry

Chapter 10

Radiometry and photometry

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Why radiometry and photometry? The need to quantify light Radiometry of point sources Radiometry of surfaces – irradiance Extended sources – radiance From radiometry to photometry Radiometry and photometry of specific optical systems Radiometers and photometers Spectroradiometry and spectrophotometry Measurement of color

10.1 Why radiometry and photometry? The need to quantify light Previous chapters have concentrated on the propagation of light and predominantly the direction taken by rays as they travel through an optical system. There are also many instances when it is necessary to quantify the amount of electromagnetic radiation present. This chapter concentrates on this area and also, importantly, the measurement of light emitted by sources and traveling through optical systems. Light refers to electromagnetic radiation that elicits a response from the human visual system. The resulting range of wavelengths is referred to as the visible spectrum (see Figure 9.12). The world is dominated by a major source of light, the sun and in addition, we have developed a wide range of artificial sources to complement natural sources. As a result, our eyes and brain process a vast amount of visual information in terms of shape, contrast, movement, texture, and color in our everyday lives. The following points emphasize why there is a need to quantify electromagnetic radiation and in particular light as its visual percept: . . .

The eye is imprecise in its assessment of quantities such as light level; it is unlikely that two observers will agree on this aspect of a scene. Excessive radiation levels can cause temporary and sometimes permanent damage to the eye and other exposed tissues. Light sources and other radiation equipment need to be tested accordingly. Light sources need to be calibrated and rated for their output so that different products can be compared. 344

10.2 Radiometry of point sources

. .

345

Lighting levels need to be sufficient to create a safe and comfortable working environment. Lighting designers need to be able to specify sources and predict their effect within a particular environment.

So what are radiometry and photometry? Radiometry is the science of measurement of electromagnetic radiation and photometry is a subset of radiometry where the measurements are related to the visual response of the eye under normal light adapted (photopic) and, less usually, dark adapted (scotopic) conditions. A photometric measurement of ultraviolet radiation would give a near zero value since the eye is not sensitive to these wavelengths. However, a radiometric measurement of the level of ultraviolet radiation is important since such short wavelengths (notably the UV-B waveband) are potentially harmful. Everyday examples include snow blindness and welder’s eye, two conditions brought about by exposure to ultraviolet radiation. There is also a well-established link between skin cancer and exposure to the UV-B band of wavelengths. Beyond the other end of the visible spectrum, infrared wavelengths that are invisible can be used to cause retinal burns for therapeutic uses. Measuring and thereby controlling levels of infrared radiation has safety implications in certain applications.

10.2 Radiometry of point sources A point source is an idealized source whose dimensions are very small compared with the viewing distance. Specifically the product of the lateral source dimensions should be very much smaller than the distance to the detector or observation screen squared and hence the solid angle subtended by the source (see below for definition of solid angle) is very small. We may therefore write our definition of a point source as dA dB  r 2

ð10:1Þ

where dA and dB are the lateral source dimensions and r is the distance to the observation screen. Although an idealization, there are many practical situations where physically large sources may be considered to be point sources if the observation distance is large enough. For example, stars can be many times the earth’s diameter but may still be considered as point sources. We shall first consider the radiometry of a point source and then extend the concepts to extended sources. A point source will emit a certain amount of radiant energy over time. The symbol widely accepted for radiant energy is Qe and it has units of joules. The total energy emitted by a source is one of its fundamental quantities. Radiant energy is particularly important when the output is not constant, for example in pulsed lasers, or where the total dose received needs to be quantified. This latter category includes some photobiology experiments and situations where the potential damaging effect of the radiation is cumulative. The radiant flux, or power of the source, denoted by the symbol fe , is the amount of energy radiated per unit time and therefore has units of joules per second (J s1) or watts (W). Radiant flux is more commonly used than radiant energy since most sources produce an approximately constant output. Returning to the example of the pulsed laser, it should now be apparent that it is possible to have a low radiant

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Figure 10.1 An isotropic point source radiates equally in all directions

Figure 10.2 Definition of solid angle 

energy but high pulse power and that the pulse power will increase as the pulse duration is decreased. The radiant energy emitted by an idealized point source is equal in all directions and hence the source is referred to as isotropic (Figure 10.1). The radiant energy is therefore emitted into a sphere. It is rarely possible to collect light over an entire spherical surface since most detectors only have a small active area. It is therefore important to define how much of the spherical surface is being measured and this is achieved using the concept of a solid angle. A solid angle defines the collection cone of the detector with the source at its apex and is an extension of the more familiar scalar angle into three dimensions. A solid angle is defined as the ratio of the area on the surface of a sphere centered on the source point to the radius of the sphere squared. It is commonly given the symbol  (Figure 10.2). Our idealized point source, emitting radiant energy in all directions, irradiates an area 4pr2 of the sphere and therefore emits into a solid angle of 4p steradians, where steradians are the units of solid angle. If, however, a detector of area A is placed a distance r from the point source then the detector subtends a solid angle given by ¼

A r2

ð10:2Þ

at the source. Most practical sources do not radiate uniformly in all directions and therefore we define the radiant intensity of a source as the radiant flux per unit solid angle in a given direction. It is given the symbol Ie and has units of watts per steradian (W sr1). Radiant intensity can therefore by expressed by Ie ¼

fe 

ð10:3Þ

where the symbols have the same meaning as before. Figure 10.3 illustrates a situation where two detectors of different active area subtend the same solid angle at a point source and therefore collect the same amount of radiant flux. If a detector of active area A is placed at different distances from the source then less radiant flux is collected but over a smaller solid angle and hence the radiant intensity will be the same since it measures the radiant flux per unit solid angle (Figure 10.4).

10.3 Radiometry of surfaces – irradiance

347

Figure 10.3 The solid angle subtended by detector of area A1 is the same as a larger detector of area A2 located further from the source

Figure 10.4 Although a detector of area A collects less light as it is moved away from the source the measured intensity remains constant

10.3 Radiometry of surfaces – irradiance An idealized point source emits equally in all directions and therefore has a radiant intensity that is constant for all directions. The radiant flux emitted into a given solid angle is therefore independent of direction. If a screen of area A is placed a distance r from the source then the solid angle subtended is A/r2 steradians. Moving the screen further away reduces the solid angle and hence the screen receives less radiant flux. A measure of the radiant flux incident on the screen (or a detector) is the irradiance defined as the radiant flux per unit area of the screen or detector. It is given the symbol Ee and has units of W m2. Irradiance can therefore be expressed by fe ð10:4Þ A where the symbols have the same meaning as defined above. Substituting for fe from Equation 10.3 and for A from Equation 10.2 yields Ee ¼

le ð10:5Þ r2 which is the well-known inverse square law. This equation tells us that the irradiance varies as the distance between the source and receiving surface is changed (Figure 10.5). However, Equation 10.5 predicts the irradiance at only the foot of the perpendicular from the source to the screen. For a point source the irradiance is constant over a spherical surface. This follows from the definition of a solid angle. However, most detectors and observation screens are flat and hence, in general, the irradiance will not be constant across the receiving surface. The change in irradiance is due to the geometry since most of the light is no

Ee ¼

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Radiometry and photometry

e

Figure 10.5 The inverse square law of irradiance

longer normally incident on the receiving surface and also because the distance from the source to the screen changes. We shall now consider these two factors to determine how the radiance changes across a plane surface when irradiated by a point source. A small element A on a sphere is the same size when viewed from the source no matter where it lies on the sphere since it is always normal to the direction of propagation from the source. If this small element is projected onto a flat surface at point P (Figure 10.6) then it will cover a larger area A 0 that is given by A ð10:6Þ cos y where y is the angle between the direction of propagation from the source to point P and the perpendicular to the screen passing through the source. From Equation 10.4 the irradiance at P can be found by replacing the area of the screen A by the expression in Equation 10.6 giving A0 ¼

E0e ¼ Ee cos y

ð10:7Þ

where Ee is the irradiance at the foot of the perpendicular to the screen passing through the source. The distance from the source to point P is also increased such that r ð10:8Þ r0 ¼ cos y From Equation 10.5, after replacing r by r0 given in Equation 10.8, we obtain E0e ¼ Ee cos3 y

ð10:9Þ

10.4 Extended sources – radiance

349

z

Figure 10.6 Variation in irradiance when a point source irradiates a screen

which is known as the cos3 law of radiometry and photometry. This result shows that the irradiance of a plane surface by an isotropic point source falls off from a maximum at the foot of the perpendicular to the screen passing though the source as cos3 y (Figure 10.6).

10.4 Extended sources – radiance Although there are a large number of situations where we can approximate a source as a point source there are others where this approximation is not valid. For example, as soon as we use a point source to irradiate an observation screen then the screen becomes an extended source that could, in turn, produce a response from a detector such as our eyes. One possible way to analyze this problem is to consider an extended source as a collection of point sources, an approach suggested by O’Shea in Elements of Modern Optical Design (O’Shea 1985). Such an approach has the advantage of building on what we already know and is intuitive. For an extended source, the concept of radiant intensity needs to be extended to account for the source area and the fact that the radiant intensity may vary across this area. The radiance is the radiant flux emitted into unit solid angle per unit area of source in a given direction. It therefore has units of watts per square meter per steradian and is given the symbol Le. If we consider an extended source comprising equally spaced point sources then Le ¼

fe A

ð10:10Þ

where fe is the radiant flux of the source, which has an area A, and  is the solid angle subtended by the detector at the source. If the same extended source is now viewed from an angle y then the point source spacing in one direction is foreshortened and the value of A is reduced by a factor cos y (Figure 10.7). Replacing A by A cos y in

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Radiometry and photometry

Figure 10.7 An extended source considered as a collection of point sources viewed (a) at normal irradiance and (b) from an angle y

Equation 10.10, the radiance at an angle y becomes Le ðyÞ ¼

Le ð0Þ cos y

ð10:11Þ

where Le(0) is the radiance along a direction normally incident to the source surface. Unfortunately this equation implies that the source radiance increases and would be infinite at grazing incidence, a fact that is readily dismissed by simple observation. In fact the brightness of most extended sources or surfaces usually decreases or remains roughly constant over a wide range of viewing angles. One possibility is to consider a real extended source whose radiance is independent of viewing angle. This is achieved if the radiant intensity of the point source falls off as cos y thus compensating for the inverse cos y increase seen in Equation 10.11. The radiant intensity therefore becomes le ðyÞ ¼

fe cos y 

ð10:12Þ

and the source radiance can then be shown to be independent of direction since Le ðyÞ ¼ Le ð0Þ

ð10:13Þ

A source that satisfies this condition is known as a Lambertian radiator (Figure 10.8). Although most practical sources do not meet this condition, many do over a range of viewing angles making it a very useful approximation. We can now calculate the irradiance on a screen placed a distance r from the extended source. The irradiance produced by a small element dA of the source is given by Equation 10.10. After substituting into Equation 10.3 the radiant intensity is given by le ¼

fe ¼ Le dA 

ð10:14Þ

and substituting into 10.5 gives Ee ¼

Le A ¼ Le  r2

ð10:15Þ

10.4 Extended sources – radiance

351

Figure 10.8 A Lambertian source has an intensity that reduces as the cosine of the angle between the normal and the direction of propagation

where dA has been replaced by the area A, of the source, since we need to sum all elements to cover the whole source area.  is simply the angle subtended by the detector/observation screen at the source. Radiance is conserved in linear optical systems where there are negligible transmission losses. This can be seen in the example of a positive lens imaging an extended source. It would initially appear that the lens concentrates the light. However, to produce a smaller more concentrated image of the source, the object conjugate must be large and therefore the lens subtends a smaller solid angle at the source and collects less light. If the lens is moved closer to the source to subtend a larger angle and collect more light, the image conjugate increases and the source area is imaged over a larger image area. Although a formal proof of this theorem is beyond the mathematical scope of this text, a limited explanation is given below. Consider a single positive thin lens, located a distance l from extended source, that forms an image of the extended source on an observation screen located a distance l 0 from the lens (Figure 10.9). The solid angle subtended by the lens at the source is therefore given by  ¼ A/l2 and hence the total radiant flux colleted by the lens is given from Equation 10.10 by A ð10:16Þ l2 where the usual subscript, e, for radiometric quantities has been dropped in favor of using a subscript S for source quantitites and D for detector or observation screen f ¼ AS LS

Figure 10.9 Lens imaging an extended source – conservation of radiance

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Radiometry and photometry

parameters. It is logical to determine a radiance for the observation screen since this in turn becomes the source for another optical system or detector. We first note that the solid angle subtended by the lens aperture at the screen is given by 0 ¼ A=l 0 2 and that the area of the image is related to the source area by AD ¼ M2 AS where M is the transverse magnification. The radiance of the observation screen is therefore given by: LD ¼

f 0 A

D

ð10:17Þ

which after substituting for the quantities in the numerator and denominator on the right-hand-side and tidying yields LD ¼ LS

ð10:18Þ

hence showing that the radiance is conserved. The above result can be generalized to complex optical systems by replacing the lens aperture with the entrance pupil and exit pupil of the optical system. A more formal proof involving the e´tendue ge´ometrique is given on pages 44–46 of Handbook of Applied Photometry (DeCusatis, 1997). It should be noted that this is the radiance of the aerial image for light traveling in the direction of propagation. In a large number of situations, the radiant energy is reflected by a diffusing screen, which is then viewed by an observer. This problem is different and will be considered in Section 10.6 when we consider the radiometry of a slide projection system. Although the radiance does not change, a lens imaging an extended source will, in general, alter the irradiance at the detector. Using a magnifying glass to start a fire provides a simple example; this is the same principle that Archimedes, fanciful legend has it, employed, although with a mirror, for setting fire to the Roman fleet at Syracuse. In both cases (lens and mirror), a minified image of the sun is formed in the focal plane. If the flux density (irradiance) is sufficient then the heating effect could ignite or damage any material placed at the lens focus. This is the basis of photocoagulation used to seal leaking retinal blood vessels where opthalmologists use lasers to produce a very small, localized heating effect that can seal the aneurysm. An expression for the change in irradiance at the detector is obtained as follows: the irradiance at the detector/observation screen is given by ED ¼

f AD

ð10:19Þ

where f is the radiant flux collected by the lens. Substituting from Equation 10.16 the detector irradiance is therefore given by ED ¼

AS LS  AD

ð10:20Þ

Recalling that the product LS is simply the source irradiance and the ratio AD/AS is the transverse magnification squared, finally yields ED ¼

ES M2

ð10:21Þ

Systems that minify objects will therefore increase the image irradiance whereas those that magnify objects produce a reduced image irradiance. An example that illustrates this effect well is a 35 mm slide projector. A simple optical layout for this system is given in Figure 10.10 with typical values for the magnification of 30 to 50.

10.4 Extended sources – radiance

353

Figure 10.10 Simple slide projector arrangement with Ko¨hler illumination (Source imaged into pupil of projection lens)

We can usefully rewrite Equation 10.21 using the plane angle u since this relates to the numerical aperture and hence the f-number of the projection lens. Since the marginal ray slope angle u is given by u = D/2l where D is the lens entrance pupil diameter and the solid angle subtended by the lens at the source is  ¼ pD2 =4l 2 it follows that  ¼ pu2 . Hence substituting into Equation 10.21 it follows that pu2 LS ð10:22Þ M2 and hence it is possible to get a higher screen irradiance by increasing the numerical aperture of the projection lens. This can be achieved by reducing the focal length or by expanding the entrance pupil diameter. However, both approaches will place increased demands on the aberration correction of the lens. More sophisticated illumination systems are therefore often used. When a point source irradiated a plane surface, we previously found that the irradiance varied across the screen as cos3 y (Equation 10.9) where y is the angular position on the screen measured from the normal to the screen that passes through the source point. It is again important to know what the variation in irradiance is across the screen when irradiated by an extended source since this encompasses most practical situations such as the illumination in the film plane of a camera. For our starting point we will assume that a lens of area A is imaging the source onto a screen. The flux collected by the lens for a source point on-axis is given by Equation 10.16. Off-axis, the distance to the lens is increased by a factor 1/cos y and the lens area projected onto a plane perpendicular to the direction of interest is reduced by a factor cos y as has already been argued when a point source irradiates a plane observation screen. In addition, for an extended source, the source area projected onto a plane perpendicular to the direction of interest is also reduced by a factor cos y and hence the flux collected by the lens is given by ED ¼

fe ðyÞ ¼ AS LS  cos4 y

ð10:23Þ

where  is the solid angle subtended by the lens at the axial object point on the source. From Equation 10.4 and 10.23, the irradiance ED at the observation screen, which has area AD, is therefore given by ED ¼ LS 0 cos4 y

ð10:24Þ 0

since it can be shown that  , the solid angle subtended by the lens at the observation screen, is given by 0 ¼ M2 . It is left as an exercise for the reader to show that this

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Radiometry and photometry

is the same result as would be obtained if a source of radiance ES and area AS were located at the lens itself. This last result is of particular importance in a large number of imaging systems where we can consider the exit pupil of the optical system as the extended source irradiating an observation screen or detector. In a camera, for example, Equation 10.24 produces an upper limit to the fall off in irradiance away from the optical axis even if the source is Lambertian. Vignetting, where the field stop is not located at an image, will cause an even greater fall off in iradiance. This is because part of the solid angle is occluded reducing further the light flux reaching the observation screen. Vignetting is covered in Section 5.8 of this book; its effect on image plane irradiance is further explained in Section 3.31 of O’Shea (1985). Although there are some clever tricks with aberrations that attempt to compensate for the cos4 y fall off in irradiance, it is often difficult to beat and hence you will commonly see photographs, notably from wide-angle lenses, that get darker towards the corners of the print. There are a number of situations where a physical source may be approximated by a point source. The question is therefore when is this approximation valid? We can gain an initial insight into this by returning to Equation 10.1 and noting that this specifies that the solid angle subtended by the source at the detector is very much less than 1 steradian. However, we can now approach the problem more formally by comparing the results of using a point source approximation for a disk shaped extended source. For a point source, the inverse square law applies and hence from Equation 10.5 the irradiance, EP, can be written as pa2 L ð10:25Þ r2 where r is the detector distance, a is the source radius, and a subscript P has been used to indicate that this result is for a point source. For an extended disk source of radius a, the irradiance produced on a screen can be shown to be given by EP ¼

EE ¼

pa2 L a2 þ r2

ð10:26Þ

where the symbols have the same meaning as before and the subscript E has been used since it is the result for an extended source. If we wish to have less than 1% error then EP  EE  0:01 EE

ð10:27Þ

After substituting and tidying we obtain a  0:1r

ð10:28Þ

This result is the basis for the ‘‘five times’’ rule of thumb that can be used when determining if the point source approximation holds; the rule simply states that the observation distance should be five times the largest source dimension.

10.5 From radiometry to photometry 10.5.1 The CIE standard observer It has already been indicated in Section 9.4 that different wavelengths give rise to the sensation of different colors. The white light spectrum shown in Plate 5 also

10.5 From radiometry to photometry

355

Figure 10.11 Luminous efficiency functions for photopic (solid line) and scotopic (broken line) vision

demonstrates that the sensation produced by yellowish-green light is stronger than that produced by either blue or red light. This latter point is true for a source emitting equal energies at all wavelengths present and hence our eyes have a relative spectral response that depends on the wavelengths present with a maximum in the yellow–green and a very small or zero response in the ultraviolet and infrared regions of the spectrum. The average spectral response of the human eye is known as the relative photopic luminous efficiency function. It is also often referred to as the photopic response or V(l) curve and was originally known as the visibility factor. The average value of this function has been determined for many human observers with normal color vision and hence defined as the Standard Photometric Observer (Figure 10.11). Unfortunately there is great difficulty in obtaining agreement between different observers and this is one of the main reasons for adopting the Standard Photometric Observer rather than using individuals. This lack of agreement is also the reason that methods of visual photometry (see Section 10.7) show poor reproducibility. The V(l) curve has a peak at 555 nm and all values are normalized such that the response at the wavelength is unity. The luminous efficiency of every other wavelength is therefore relative to the response at the peak of 555 nm. More specifically, for monochromatic light of wavelength l, the value of V(l) is the ratio between the radiant flux at 555 nm to that at wavelength l producing an equally intense visual sensation for foveal vision under supra-threshold conditions. When the light levels fall to less than 103 Cd m2 the mechanism of seeing changes over to rod dominated vision and the eye is said to be dark adapted. Under

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these scotopic conditions the photopic response curve is no longer valid. The relative scotopic luminous efficiency curve, or V0 (l) function, is also shown in Figure 10.11. There is a shift in spectral response towards the blue end of the spectrum, the socalled Purkinje shift named after the famous Czech physiologist who was the first to observe that red colors fade more quickly than blue as the ambient illumination falls. The scotopic luminous efficiency function is specific to young observers (less than 30 years of age) and for observations made at least 5 degrees away from the fixation under complete dark adaptation. The eccentric fixation is necessary because of the absence of rods at the foveola. 10.5.2 Photometric quantities Using the ideas introduced in the previous section, it is possible to extend our work on radiometry to cover the measurement of light – photometry. Here, we take light to mean any incident radiation capable of generating a visual percept directly. All of our radiometric quantities can therefore be turned into photometric quantities by weighting the radiometric measurement by the photopic response curve, provided that the observer is light adapted and using the center of their vision (a 2 field is recommended). For example, luminous flux, fv , that is the radiant flux that generates a response from the average eye, is given by Z 830 fe ðÞVðÞ d ð10:29Þ fv ¼ 683 360

where a subscript v is used on photometric quantities whereas e has been used for the corresponding radiometric quantity. The radiant flux varies with wavelength and hence we have written it as fe ðÞ to indicate that its value at wavelength l is to be R used. The symbol is a mathematical integral; it simply means that for each value of the wavelength (in theory, in infinitesimally small steps but in practice 10 nm steps are often sufficient for calculations), the radiant flux, fe ðÞ, at wavelength l is multiplied by the V(l) function at the same wavelength and all the values summed. In fact, for those unfamiliar with calculus, the integral sign can be usefully thought of as a horizontally compressed ‘‘S’’ standing for ‘‘sum’’. Equation 10.29 implies that the luminances at individual wavelengths can be added to produce the overall response of the eye, the so-called linearity condition. Although there are nonlinearities in the visual system equation 10.29 is a good approximation under most conditions. An example where luminances are not additive is for saturated (near monochromatic) colors. This is known as the Helmholtz–Kohlrausch effect where lights with saturated colors appear brighter than a white light that has the same luminance. Everyday examples of the Helmholtz–Kohlrausch effect include traffic lights and rear lights on cars. In these cases care must be taken in predicting the response of the eye owing to this nonlinearity. However, most reflected colors have a low saturation (a mixture of the pure spectral color and white) and the linearity condition holds well. The unit of radiant flux is the watt whereas the corresponding unit of luminous flux is the lumen. 1 watt of radiant flux at 555 nm produces a luminous flux of 683 lumens hence the appearance of the constant 683 in Equation 10.29. It is well worth nothing that this constant has changed over the years. It started life in 1903 as 621 lumens watt1 becoming 673 lm W1 in 1907 then 685 lm W1 (1948) and 680 lm W1 before its current value of 683 lm W1 was adopted in 1980. When

10.5 From radiometry to photometry

357

looking at examples or calculations in older texts it would be sensible to check the value of this ‘‘constant’’. A more fundamental photometric quantity is luminous intensity, Iv, which has units of candelas. The candela is a base unit in the SI system hence its importance. By definition a candela is the luminous intensity in a given direction emitted by a monochromatic source of frequency 540  1012 Hz that has a radiant intensity in that direction of 1/683 Wsr1. Specifying the source in terms of frequency allows the definition to be applied in different media. For standard air 540  1012 Hz corresponds to a wavelength of 555.016 nm. The luminance is the luminous intensity emitted per unit area of source and therefore has units of cd m2. It is given the symbol Lv and is closely associated with the visual perception of brightness. As with radiance it is conserved in optical systems where there are negligible transmission losses. This is sometimes referred to as the brightness theorem. Source luminance values vary considerably; Table 10.1 gives typical values observed for common sources. The apparent anomaly, where a fluorescent lamp has a lower luminance than a candle, comes from the consideration of the source geometry; fluorescent lamps emit light from the phosphor coating on the tube over its entire surface whereas tungsten filaments are much smaller. The illuminance is the number of lumens per unit area falling on a surface. Illuminance is given the symbol Ev and has units of lux where 1 lx equals 1 lm m2. As with the luminance of sources, the illuminance values can vary over a large range. This has implications for illuminance meters discussed in section 10.7.2. Table 10.2

Table 10.1 Typical luminance values for common sources of light

Source

Sun Arc lamp Tungsten filament lamp Candle Fluorescent tube Moon Clear blue sky Overcast sky Moonless clear night sky

Luminance ðcd m2 Þ 1.6  109 3.0  108 2.0  106 to 3.0  107 2.0  106 1.0  104 2.6  103 1.0  104 1.0  103 1.0  103

Table 10.2 Typical illuminance values proposed by differing sky conditions

Source

Illuminance ðlxÞ

Direct sunlight Clear daylight Overcast daylight Full moon Moonless clear night sky

1.0  105 1.0  104 1.0  103 1.0  101 1.0  103

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Table 10.3 Summary of radiometric and photometric units and their generally accepted symbols

Quantity

Unit

Symbol

Radiance energy Radiant flux Radiant intensity Radiance

Joule

Qe

Watt

fe

Watts per steradian Watts per square meter per steradian

le

Watts per square meter

Irradiance

Quantity

Unit

Symbol

Talbot

Qv

Lumen

fv

Ee

Illuminance

Candela (Lumens per steradian) Candelas per square meter (Lumens per square meter per steradian) Lux (Lumens per square meter)

lv

Le

Luminous energy Luminous flux Luminous intensity Luminance

Lv

Ev

gives an indication of the range of illuminance values produced by a number of sky conditions. As can be seen, the range is from 0.001lx to 100 000 lx, a range of 8 log units. To complete this section, Table 10.3 summarizes the various radiometric and their corresponding photometric quantities discussed in the text. It should be noted that the areas referred to for irradiance/illuminance are the illuminated areas of the observation screen/detector whereas the area referred to in the units of radiance/ luminance is a source area. The concept of intensity should only be applied to a point source whereas radiance/luminance applies to extended sources, which can be approximated as point sources under the conditions outlined above. The quantities listed above and their units are part of an international standard. Unfortunately both radiometry and perhaps to a greater extent photometry have used a large number of other units in the past. Conversion factors can be found, if needed, in the article by Roberts (1987). 10.5.3 Photometric standards The earliest standards for light were candles and comparisons were made between a standard candle and other sources. This gave rise to ‘‘candle power’’ as the unit of luminous intensity. Not surprisingly candles were not reproducible to the accuracy required and so from the mid nineteenth century a series of flame lamps such as pentane lamps (Vernon–Harcourt lamp) and the Heffner lamp were developed as standards of luminous intensity. Although much effort was expended on manufacturing details and numerous correction factors, none of the flame standards proved adequate. Towards the end of the nineteenth century it was also found that the emission from a specified area of a material (platinum) at a specified temperature, the so-called Violle standard, was also too variable and that manufacturing tolerances for suitable incandescent lamps to act as standards were unachievable. From the early twentieth century onwards efforts were made to develop the use of blackbody radiators as standards. The melting point of platinum was chosen and this was adopted as the standard for defining the candela, the successor to the candle, and was defined by the Confe´rence Ge´ne´rale des Poids et Mesures (CGPM) in 1948. This standard remained in force until 1979 when the latest definition of the candela was agreed (see Section 10.5.2). The advantage of the current definition, which does not specify a particular source, is that it gives freedom for the various national standards

10.6 Radiometry and photometry of specific optical systems

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laboratories, such as the National Physical Laboratory (NPL) in the UK and the National Institute of Standards and Technology (NIST) in the USA, to come up with innovative and possibly better methods of determining the standard. Most standards laboratories realize their primary standard of luminous intensity with absolute radiometry. This involves measuring the absolute responsivity of a detector using one of several methods that go beyond the scope of this text. It is worth noting, however, that the uncertainty in such measurements is 0.2–0.4%. What has been discussed so far relates to realizing the primary standard of luminous intensity within standards laboratories of individual countries. There is obviously a need for what are commonly termed secondary or transfer standards that can be used within industrial and research laboratories. These are mostly achieved using standard lamps. In general, luminous intensity standard lamps are specially manufactured incandescent or quartz–halogen lamps. The lamps are often very expensive and typically have large gas filled envelopes to reduce blackening and improve the aging characteristics. Although quartz–halogen lamps tend not to be very stable, they are being increasingly used as photometric standards due to their low aging rate, compact size, and low cost. V(l) corrected detectors (photometers) can also be good enough to act as transfer standards. In fact this is preferable since they tend to not have the susceptibility to mechanical shock, aging, and drift during stabilization common with standard lamps. Long-term stability should be about 0.1% per year although it should be noted that some photometers exhibit changes of 1% per year making them unsuitable as secondary standards. A stable photometer only acts as a secondary standard of illuminance; secondary luminous intensity standards can only be realized with accurate measurement of distance.

10.6 Radiometry and photometry of specific optical systems 10.6.1 The human eye – retinal illuminance and the troland In common with most detectors, the eye responds to the flux density (the illuminance) at the detector where the illuminance is the photometric equivalent of irradiance (see Section 10.5 for definition). If we consider an eye viewing an extended source of luminance Lv located at a distance l from the eye, then the total flux collected by the eye is given by fv ¼

Lv AP l2

ð10:30Þ

where A is the source area and P is the area of the entrance pupil of the eye (Figure 10.12). This luminous flux is now spread over an area A0 of the retina, which is located at a distance l 0 from the exit pupil, and hence the retinal illuminance is given by  0 2 n Ev ¼ Lv P 0 ð10:31Þ l after recalling that A0 = M2 A and that M2 ¼ ðn=l Þ2 ðl 0 =n 0 Þ2 . M is the transverse magnification, l and l 0 are the object and image conjugates respectively and n and n0 the object and image space refractive indices. Equation 10.31 is only approximate

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Figure 10.12 Retinal illuminance and the troland

since it also depends on absorption and scattering within the eye and also the efficiency of absorption of the incident light by the cone photopigments (the Stiles– Crawford effect). In addition, parameters such as n 0 and l 0 are often unknown although in most practical situations it is reasonable to approximate l 0 with the second equivalent focal length of the eye. To avoid assumptions, a measure of retinal illuminance can be given by the quantity LvP, which is proportional to the true retinal illuminance. The unit of retinal illuminance defined for this quantity is the troland, which is defined as the retinal illuminance produced when an eye with a pupil of area 1 mm2 views a uniform extended source of luminance 1 cd m2 Therefore for a pupil diameter of 3.57 mm viewing a source of luminance 1 cd m2, the retinal illuminance is 10 trolands. Values of retinal illuminance can get quite large and so log trolands are often used. This also makes sense since the eye responds logarithmically to illuminance. As a final example, the luminance of the sun is approximately 1.6  109 cd m2. If the pupil diameter is 1 mm then the retinal illuminance is 9.1 log trolands – a dangerously high level that can cause retinal burns. Viewing the sun directly or through optical instruments such as binoculars or cameras should never be attempted; indirect methods of viewing should be used when observing, for example, a solar eclipse. 10.6.2 Slide projector We have seen in the previous section that the radiance is conserved in a linear optical system (Equation 10.18). Linear systems include all conventional imaging systems but exclude those with image intensifiers, photomultiplier tubes, lasers, or other devices that utilize nonlinear phenomena. However, a slide projector produces a good example where careful consideration of the radiometric aspects produces a different result to those obtained so far. The crucial factor here is to consider the way that light is reflected by the diffusing (projection) screen (Figure 10.13). In many cases this screen will act as a Lambertian diffuser and it can be shown that a Lambertian source of size dA produces a total radiant flux of pLe dA where Le is the radiance of the source into a hemisphere. We can use this result to find the radiance of the diffusing screen in our slide projector example on the assumption that the screen is Lambertian. From Equation 10.21 the irradiance at the diffusing screen is given by ED ¼

pu2 Ls M2

ð10:32Þ

where u is the marginal ray slope angle at the projection lens and again the subscript S is used to indicate a source parameter and subscript D a parameter at the diffusing screen. A small element dA of the screen emits a total flux EDdA into a hemisphere

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Figure 10.13 Luminance of a Lambertian diffuser illuminated by an extended source

since the screen is a perfect Lambertian diffuser. This total flux can be equated with the expression introduced in the previous paragraph to finally yield L0 ¼

u2 LS M2

ð10:33Þ

where L0 is the reflected radiance of the diffusing screen. The photometric equivalent of radiance, luminance, corresponds to perceived brightness and so there is usually considerable loss of brightness in projection systems because of the large value of the magnification, M. This can be compensated for with a large collection angle for the projection lens if a high screen brightness is required. As with all such results, any losses due to absorption, reflection, or scattering in the system can be accounted for by multiplying the final result by a transmission factor. If the screen is not a perfect Lambertian diffuser then it may appear brighter from certain directions that others. This may be acceptable, for example, when projecting slides in a lecture theater or cinema, since observers do not tend to sit at very oblique angles to the screen. The ratio of the reflected luminance from a screen to that from a perfect Lambertian diffuser in the same direction is known as the gain of the screen. Retro-reflective materials, such as those developed for use in road signs and car number plates where the driver sits within a few degrees of the direction of the headlights, may have gains of 1000% whereas matt white paint may be little more than 100%.

10.7 Radiometers and photometers 10.7.1 Visual photometers There are seven internationally agreed base units of physical measurement in the SI System. Only one of these units, the candela, is physiologically based. It will therefore be useful to consider subjective photometers both from a historical perspective and

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from the difficulties in executing accurate and reproducible photometric measurements with a human observer. It has been noted at the start of this chapter that the eye is extremely poor at making absolute judgments on the amount of light present. However, it is much better at assessing relative differences and hence comparison photometers almost invariably compare the luminance of a test field with a comparison field. The smallest increment in brightness between two fields that can be detected by the human eye is given by the Fechner fraction, named after Gustav Fechner (1801–1887) who formalized Weber’s law. Under a wide range of conditions Fechner’s fraction is about 1%. In crude implementations, the test and comparison photometer fields would be created by two translucent diffusing schemes, one receiving light from the light source to be measured and the other from a comparison lamp. The distance of the test lamp or the comparison lamp from their respective screens is altered to balance the luminosities of the two fields via the inverse square law. Such an approach is prone to many errors and the following conditions should be met if a reasonable degree of accuracy is required: 1. 2. 3. 4.

Adjacent test and comparison fields Photopic conditions A small field size (5–7 recommended) Light from the test and comparison lamps must fall at right angles onto the diffusing screens 5. 20–100 lm m2 at the diffusing screens is recommended for direct comparison

A device that meets most of these requirements and is still seen within laboratories for demonstration purposes is the Lummer–Brodhun comparison photometer (Figure 10.14). At the heart of this instrument is a pair of Lummer & Brodhun

Figure 10.14 Lummer–Brodhun photometer head with equality of brightness field illustrated

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prisms first developed in 1889. These are two right-angled prisms with their hypotenuse faces opposed. The hypotenuse face of one of the prisms has a spherical surface polished on it with the exception of a small central area that is left flat. The effect is to allow light incident on one prism to pass through the central field and light from the other prism to pass through the peripheral area. This produces an ‘‘equality of brightness’’ field as shown in Figure 10.14. The advantage of this arrangement is that there is no significant diffraction at the boundary between the test and surround field. This is very important since diffraction would allow the boundary to be seen even when there was equality of brightness. The light from each source passes through an aperture in the metal case and falls on either side of a white diffusing screen S; this is formed of plaster of Paris or of two sheets of ground white opal glass with a plate of metal between them. Two rightangled totally reflecting prisms P1 and P2 reflect the diffused light from the screen surfaces to the Lummer–Brodhun cube P3P4, Light from the left-hand side of the screen reaches the eye via the central part of P3P4 while the surrounding area receives light from the right-hand side. The entire field will appear uniform when a balance is obtained. A modification of this comparison photometer to give greater sensitivity uses the Lummer–Brodhun contrast cube. The opposing hypotenuse faces of the two Lummer & Brodhun prisms are modified such that they produce two hemi-fields; the test field is superimposed on the comparison field in one hemi-field whereas the comparison field is superimposed on the test field in the other (Figure 10.15). By reducing the luminance of one of the surrounding hemi-fields and one of the trapezoids using glass plates attached to the prisms, the task is then for an observer to balance the contrast (about a 5% difference is usual) between each trapezoid and its corresponding hemi-field. When making measurements using such devices, a substitution method is often employed to reduce errors due to a difference in the diffusing screens. In this technique a comparison lamp of similar spectral content to the test lamp is used to create the comparison test field. The fields are first balanced for the test lamp and

Figure 10.15 Lummer and Brodhun equality of contrast prisms and resulting photometer field, Glass plates G1 and G2 are used to reduce the luminance of the trapezoids

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then for the standard lamp. The test lamp has been substituted with the standard lamp hence the name for the technique. Chromatic difference between the test and comparison field upset luminosity matches. Flicker photometry is a method that can be used in this situation and works by alternating the comparison and test fields in the field of view of the observer. It is based on the fact that the eye responds more slowly to color stimuli than to luminance stimuli. As the flicker frequency is increased the eye perceives changes in luminance and not changes in color. For so-called heterochromatic photometry the flicker technique is much more accurate than the direct comparison method. Precautions that need to be noted are: 1. Photopic levels of luminance (greater than 3 cd m2) 2. Photometer field of 2 so that only cone vision is employed 3. Careful selection of observer. Even when these conditions are carefully controlled, flicker photometry has shown poor precision and poor agreement between different laboratories. Assessing the accuracy of visual photometers is straightforward; any results obtained should conform to the relative luminous efficiency function of the eye. Achieving good accuracy is very difficult and apart from the design details mentioned above, it should be recalled that determining the Standard Photometric Observer (Section 10.5) required very careful observer selection and a number of average responses. Spectroradiometry can also be used to derive photometric quantities and this is covered in Section 10.8. 10.7.2 Physical radiometers/photometers Modern photometers/radiometers are physical instruments whose major components are the collection optics, an electronic detector, and either optical or electronic correction for the V(l) function. They offer the immediate advantage of portability, are more precise and accurate than their visual counterparts, and can display and calculate the output in different forms. Many recent instruments are based on developments in detector and computer technology over the past 10 to 15 years. A full survey of modern instruments would be inappropriate for this text and immediately out of date. We shall briefly summarize the main features of luminance and illuminance meters and give some typical values for accuracy and measurement range. Specific details should always be sought from a reputable manufacturer before purchasing a luminance or illuminance meter. Collection optics vary from cosine corrected detectors (usually a photocell with a diffuser placed in front of it), to sophisticated optics in luminance meters and photometers. Cosine corrected detectors are needed when the receiving surface has a cosine response, that is responds equally to light incident at differing angles. More sophisticated optics are required when it is necessary to measure over a small known field of view as in a luminance meter. This requires accurate alignment and should ideally be nonpolarization dependent. The detectors used are often silicon photocells although photomultiplier tubes and CCD arrays are also employed in certain more specialized instruments. It is necessary to calibrate all of these detectors to give photometric measurements. Achieving the same response as the V(l) curve is a combination of the detector’s response and an additional filter. Correction for the relative luminous efficiency function of the eye cannot be readily handled through calculation without making assumptions about

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365

the spectral content of the source being measured. However, some illuminance meters allow a limited correction for spectral content of the source with built in calibration for common sources such as tungsten lighting, daylight, or fluorescent lighting. Matching the V(l) function is very difficult and a typical value would be achieving agreement with the Standard Photometric Observer to within 8%. The match is particularly problematic in the blue and sometimes additional filters are provided to give better accuracy at this end of the spectrum. Luminance meters require more elaborate optics and typically measure over a range of about 7.5 log units with an accuracy of 4% of the reading. Illuminance meters commonly have an integrated detector with a cosine response diffuser in front. They are accurate to within 2–3% and have a measurement range of over 6 log units. The ability to match a cosine response accurately falls at extreme angles; a typical value would be an error of 5% at 60 angle of incdence. Matching the V(l) function has already been commented upon. It should be noted that a majority of illuminance meters consist of a simple silicon photocell and readout device. Calibration is performed to correct for the relative luminous efficiency of the eye and hence measurements are only applicable when measuring an identical source to that with which the meter has been calibrated. There are many more specialized instruments such as spectroradiometers, telespectroradiometers, different types of colorimeters, and photometer imaging systems that are beyond the scope of this text. The reader is referred to the more specialized texts on radiometery and photometry and to manufacturers’ data, often available on the World Wide Web, for further details.

10.8 Spectroradiometry and spectrophotometry In Equation 10.29 it was necessary to consider the radiant flux as a function of wavelength so that the radiant flux could be weighted by the V(l) function to obtain the luminous flux. If the radiant flux is plotted as a function of wavelength then we obtain the radiant flux spectrum for that particular source. Spectroradiometry and spectrophotometry measure the wavelength dependence of the various radiometric and photometric quantities introduced in the previous section. Spectrophotometry was particularly important in determining the absorption spectrum of the cone photopigments in human eyes (see Section 10.9) and where the spectral content of a source affects the appearance of an object. A common example of the latter is often observed in photography where it will be noticed that tungsten lights produce a very orange light, fluorescent tubes a green hue, and flash lamps a more natural daylight response. The photometric power spectra of these sources illustrate why this is the case (see Figures 9.17, 9.18 and 9.20). Measurement of the spectral output of a source is conceptually easy; we simply need to attach a monochromator, a device that can separate the input radiation into contribution at individual wavelengths, to a radiometer or photometer. The monochromator allows a given radiometric/photometric quantity to be measured at a specified wavelength. The dispersive element in the monochromator, usually a diffraction grating but prisms are also used, is then scanned to cover the waveband of interest. It is important to note that the power spectrum fv ðÞ, once measured, can be used to find the total luminous power output fv via Equation 10.29; it is the area under the graph of luminous flux plotted against wavelength. This will also be true for

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other photometric and radiometric quantities. Slightly less obvious is the fact that a radiometric measurement cannot be turned into its corresponding photometric quantity unless the spectral variation of the radiometric quantity is known. Spectroradiometers can be used to characterize the spectral composition of sources, surfaces, filters, and detectors. For sources, the spectral composition is measured directly by imaging the source filament onto the input aperture of a monochromator whose output is connected to a radiometer. With surfaces, the spectral reflectance is measured. The physical arrangement needed to achieve this will depend on whether specular or diffuse reflectance (using an integrating sphere attachment) is measured and on the illuminating source characteristics. The name spectrophotometer is usually reserved for spectral transmittance measurements of samples such as filters. Spectrophotometers compare the output of a dual beam system, one passing through the test sample and the other acting as a reference. They are often self-contained with their own light source but are not as adaptable as spectroradiometers. Although spectrophotometers can measure spectral transmittance and reflectance, a spectroradiometer can be configured to measure the spectral output of light sources as well as the spectral response of detectors. Finally we note that detector spectral responsivity can be measured by a spectroradiometer in a two-stage process. First a standard detector is measured whose spectral power response is known. This permits the radiant flux at each measurement wavelength to be determined. The detector under test is then measured and its spectral power response calculated as the product of the incident flux and the detector signal. Details of the practical arrangements needed for each type of measurement mentioned above go beyond the scope of this text. Further details can be found in Chapter 8 of DeCusatis (1997).

10.9 Measurement of color 10.9.1 Introduction Color forms a major part of our world and it is used in nature and by man for such important aspects of survival as indicating danger and for camouflage. In the modern world, measurement and control of color is important in many areas of work such as the paint industry, textile industry, paper industry, and in advertising to name but a few. However, color and its measurement is a complex subject depending on the illuminating source characteristics, the material itself, and the eye as detector. These variables are well known to anyone who has tried to judge a match to the color of a piece of clothing or paint. Looking at the sample in daylight and under different artificial sources can greatly change its appearance. A match might be apparent under one illuminant but look completely different under another. The rest of this section attempts to convey the very basic principles of colorimetry. Those readers who wish to know more are referred to such texts as Measuring Colour. (Hunt, 1991) and Color Science (Wyszeki and Stiles, 1982). 10.9.2 The basics of human color perception To understand color measurement we need to first briefly look at the physiological basis of human color vision. It will only be possible to present a brief overview of this area here; the subject has been extensively covered elsewhere in the vision literature.

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Figure 10.16 Spectral responses of S ( ), M (-  -) and L (- - -) cones. The solid curve is the spectral absorbance of rhodopsin found in rod photoreceptors

The cone photoreceptors in the human retina are responsible not only for the ability to discern small detail (acuity) but also for perception of color. There are three classes of cones known as S, M, and L cones. These respond, very approximately, to the short, medium, and long wavelengths in the visible spectrum. It is tempting to think of them as blue, green, and red cones although this would be incorrect. The spectral responses of S, M, and L cones are illustrated in Figure 10.16 from which it is clear that the peak responses of the different cone types are 420 nm, 534 nm, and 563 nm respectively. Although cones are smallest and have their highest density at the fovea there are differing proportions of the three cone types across the retina. For example, there are relatively few S cones at the fovea leading to an apparent color vision defect, called small field tritanopia, if a small target is used to test color vision. It would be tempting to think that the three signals from the three different cone types are then processed to produce a color image in the same way that a cathode ray tube uses red, green, and blue channel electron guns to generate a color television picture. This is the Young–Helmholtz theory of color vision originally postulated by Thomas Young following his famous color mixing experiments. Young found that most colors could be matched by differing amounts of red, green, and blue light. These three colors – red, green, and blue – are known as the additive primaries and give rise to the largest range of mixture colors. A different selection of the three primary colors is possible (the only condition is that one primary color cannot be mixed by a combination of the other two), but gives rise to a smaller range of mixture colors. It turns out that the human visual system, in common with other species with well developed color vision, has evolved more complex ways of processing color. The

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signals from the three cone types are neurally integrated in the retina to produce three output channels; one encodes luminance, a second encodes a blue–yellow color difference signal, and a third encodes a red–green color difference signal. The two color opponent channels work with positive values indicating yellowish and reddish colors respectively while negative values indicate blueish or greenish colors. Physiological experiments support so-called color opponent theory, which was first postulated by the German physiologist Elwald Hering (1834–1918). It is salutary to note that the Young–Helmholtz theory of three basic components of any color sensation has only been confirmed relatively recently using microspectrophotometry on single cone photoreceptors. 10.9.3 Colorimetry C lumens of a given color, C, can be matched in color and luminosity by different proportions of red, green, and blue as was shown by Thomas Young. A color match to C lumens of any color can therefore be written as an algebraic relationship of the form C¼RþGþB

ð10:34Þ

where the amount of the three primary colors mixed are R lumens of red, G of green, and B of blue. R, G, and B are known as the tristimulus values since they are the amounts of the three primaries that are additively mixed. The exact proportions of red, green, and blue depend on the wavelengths of the primaries chosen but it is typical that the amount of blue in a color match is small compared with the other two primary colors. Although most colors can be matched by an additive mixture of the three primaries, some cannot. For example, the closest match that can be achieved for saturated greens and blues often appears too white or desaturated. The solution to this problem is to add some red to the original color before matching it. Equation 10.34 then becomes CþR¼GþB

ð10:35Þ

and treating this as an ordinary algebraic equation we get C ¼ R þ G þ B

ð10:36Þ

One way to understand the negative addition of red is the fact that a monochromatic source will in general cause at least two different cone types to fire therefore producing an ‘‘impure’’ response. This happens only because the spectral response curves for the photoreceptors overlap (Figure 10.16). The appearance of negative values in an additive color match was the reason Helmholtz initially rejected Young’s trichromatic theory since he knew that the photoreceptors all had to produce a zero or positive response. When Helmholtz realized that the spectral sensitivities of the photoreceptors could overlap, he unreservedly accepted Young’s theory. Carrying out color matching experiments for wavelengths spanning the visible spectrum produces the three color matching functions shown in Figure 10.17. They specify the relative proportions of the three primaries needed to match a constant amount of power over a constant small wavelength interval at each point across the visible spectrum. These three color matching functions (CMFs) are commonly denoted by rðÞ, g ðÞ, and bðÞ. Negative values for the red primary in certain color matches are

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369

Figure 10.17 Colour matching functions across the visible spectrum. Note the negative region of the red primary r() curve (see text)

clearly evident from the negative region of the rðÞ function in the blue–green region of the spectrum. The actual amounts of red, green, and blue required to match a white light of constant power over the wavelength range of interest (such a source is known as the equi-energy stimulus), are quite different. The amounts of three primaries, with wavelengths of 700 nm, 546.1 nm, and 435.8 nm, needed to match 282 lumens of an equi-energy white source are typically found to be (rounding to the nearest integer) 50 lumens of red, 229 lumens of green, and 3 lumens of blue. This color match can be written as 282W ¼ 50R þ 229G þ 3B

ð10:37Þ

following the equation for an additive color match given in Equation 10.34. However, perceptually, we think of white as ‘‘equal amounts’’ of red, green, and blue since white is not biased towards any color. If we define a new set of units such that each red unit is equal to 50 lumens, each green unit 229 lumens, and each blue unit 3 lumens then we can rewrite our color matching equation as 282W ¼ 1:0R þ 1:0G þ 1:0B

ð10:38Þ

It is important to realize that this does not affect the amounts of red, green, and blue in the match; it simply expresses it in more convenient units. The obvious advantage of this change of units is that the right-hand side can immediately be seen to represent white. However, a color match would still appear white for luminances other than 282 lumens provided the proportions of the three primaries remain the same. We can separate luminance from color by dividing the left-hand side of Equation 10.38 by 282 and by dividing all terms on the right-hand side by the sum of

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the coefficients giving the unit trichromatic equation 1:0W ¼ 0:333R þ 0:333G þ 0:333B

ð10:39Þ

where the unit of white is known as 1 Trichromatic Unit. We can now generalize Equation 10.39 to any color match of a color C 1:0C ¼ r þ g þ b

ð10:40Þ

where r, g, and b are known as the chromaticity coordinates and represent the relative proportions of the three primaries in the mix. It should also be noted that r + g + b = 1 and r, g, and b are formally given by r¼

R=RW ðR=RW Þ þ ðG=GW Þ þ ðB=BW Þ

g¼

G=GW ðR=RW Þ þ ðG=GW Þ þ ðB=BW Þ

b¼

B=BW ðR=RW Þ þ ðG=GW Þ þ ðB=BW Þ

ð10:41Þ

where RW, GW, and BW are the tristimulus values for a color match to an equi-energy white stimulus. It will be seen that Equation 10.41 is simply an algebraic statement of the numerical considerations used to arrive at Equation 10.39. The chromaticity coordinates can be plotted on a standard x–y Cartesian axis system since their sum is unity. The red and green chromaticity coordinates are usually plotted and the result is the r, g chromaticity diagram shown in Figure 10.18.

Figure 10.18 r, g chromaticity diagram with the locations of the three primaries R, G and B indicated

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The horseshoe shape is the locus of all points representing matches to monochromatic stimuli and is known as the spectral locus. Colors become less saturated as they move away from the spectral locus towards white at (0.333, 0.333). The line joining the ends of the spectrum (360 nm and 830 nm) is known as the purple boundary and hence the tongue-shaped area confined by the spectral locus and purple boundary is a color space since it contains all possible colors. One of the difficulties with the r, g chromaticity diagram is that there will always be colors that lie outside the triangle joining any three primaries chosen. Here the wavelengths of the three primaries are 435.8 nm, 546.1 nm, and 700 nm and it is clear that realizable colors fall outside the triangle joining these points. In addition, there will be negative values needed for matches to some saturated colors for reasons already discussed. In 1931 the Commission Internationale de l’E´clairage (CIE, the International Commission on Illumination) was set up and recommended a system of colorimetry that avoided this problem. The CIE achieved their goal by selecting a new set of tristimulus values, denoted X, Y, and Z, with a simple linear transformation between the two systems given by X ¼ 0:49R þ 0:31G þ 0:20R Y ¼ 0:17697R þ 0:81240G þ 0:01063B Z ¼ 0:00R þ 0:01G þ 0:99B

Figure 10.19 X, Y Chromaticity diagram

ð10:42Þ

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This transformation not only makes the new tristimulus values positive for all colors but it should also be noted that the sum of the coefficients for each equation is unity. For a white stimulus with equal proportions of R, G, and B, the relative proportions of the new tristimulus values X, Y, and Z are also equal. A final point worth noting is that the coefficients for the transformation of the R, G, and B tristimulus values to the new Y stimulus value have been carefully chosen to be in the same ratio as the number of lumens of R, G, and B needed to match an equi-energy white source. From 10.37 we would calculate these values to be 0.1773, 0.8121, and 0.0106 respectively confirming that these ratios have been used. As a consequence, the Y stimulus value is proportional to the luminance of the color. For the new tristimulus values X, Y, and Z, the corresponding chromaticity coordinates are denoted by x, y, and z. An x,y chromaticity diagram is shown in Figure 10.19 confirming that all values are positive. The only theoretical difficulty with the transformation from RGB to XYZ tristimulus values is that they represent a different choice of the three primaries. This would not be difficult to understand (we have already argued that there is a large choice of possible primaries) but for the fact that the new primaries are not real colors. However, if we accept that there are simple mathematical operations to take us from RGB tristimulus values to XYZ and any other tristimulus values that may be advantageous, then hopefully this conceptual difficulty is easier to accept. Other color spaces have been developed subsequently to correct for perceptual nonuniformity in detecting color change or for different applications within colorimetry. It is beyond the scope of this text to pursue this subject further and the interested reader is referred to one of texts mentioned in the introduction to this section.

References DeCusatis, C. (editor) (1997) Handbook of Applied Photometry. Berlin: Springer. Hunt, R.W.G. (1991) Measuring Colour, 2nd edn. Chichester, UK: Ellis Horwood. O’Shea, D. (1985) Elements of Modern Optical Design. New York: John Wiley & Sons. Roberts, D.A. (1987) Radiometry & Photometry: Lab Notes on Units. Photonics Spectra, April. Wyszeki, G., Wyszeki, G., and Stiles, W.S. (1982) Color Science. New York: Wiley.

Exercises 10.1 10.2 10.3

10.4

Think of as many specific applications as possible where accurate measurement of light level or color is required. Energy radiated by a point source of light is collected by a detector subtending a solid angle of 0.1 sr at a distance of 50 cm from the source. Find the area of the detector. A sphere of radius 10 cm has a circle of diameter 10 cm marked on its surface. Find the area of the sphere bounded by the circle and hence calculate the solid angle subtended by the circle at the center of the sphere. (The area on the surface p of a sphere of radius R bounded by a circle of radius r is given by A ¼ 2pR ½R  ðR2  r2 Þ .) Calculate the area of the circle if it is plane and again calculate the solid angle. Why is there a difference? An approximation to the solid angle is obtained by dividing the area of a lens by the distance from the source squared. A lens of diameter 4 cm is located 50 cm from a point source. Hence calculate the solid angle subtended by the lens at the source.

Exercises

10.5 10.6

10.7 10.8 10.9

10.10

10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21

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If a lens of radius r is located at a distance of 10r from a point source, show that the error between the approximate and exact values of the solid angle is 0.25%. (You will need the formula given in Exercise 10.3.) A lens of diameter 5 cm is located 30 cm from a lamp with a small filament of radiant intensity 50 W sr1. Calculate the radiant flux collected by the lens. If the lens has a focal length of 10 cm, find the angle subtended by the lens at the image and the radiant intensity at the image. Hence show that the radiant intensity of the image is simply the source intensity multiplied by the lateral magnification squared. A screen is located at a distance of 1 m from a small filament lamp of radiant intensity 25 W sr1. Calculate the irradiance at the screen. At what distance has the irradiance dropped to 10% of this value? The radiant flux from a point source of radiant intensity 100 W sr1 is collected by a lens of diameter 30 cm located 50 cm from the source. This flux is projected onto a screen of area 1 m2 located 2 m from the lens. What is the irradiance at the screen? A circular source of radiance 1000 W m2 sr1 and diameter 5 cm is located 1 m from a lens of focal length 20 cm and diameter 5 cm. By first calculating the flux collected by the lens and the area of the image of the source, find the radiance of the image and hence show that it is simply the source radiance. A small source of radiant intensity 500 W sr1 is mounted 2 m above a path. Calculate the irradiance on the path vertically below the street lamp and at distances of 1, 2, 3, and 4 m along the path from this point. Plot a graph showing the change in irradiance along the path and superimpose on this a graph of cos3 y where y is the angle between the perpendicular from the source to the path and the point being measured. Although radiance is theoretically conserved in optical systems (Equation 10.18), what practical factors will cause some light to be lost? For a single positive lens, show that the solid angle subtended by the lens at the source, , is related to the solid angle subtended by the lens at the image, 0 , by 0 ¼ =M2 where M is the lateral magnification. If the radiant flux from a source is measured, why is it not possible to calculate the luminous flux? What assumptions would have to be made to carry out this calculation? Can sources with different variations in radiant flux with wavelength, fe ðÞ, produce the same luminous flux? An eye with a pupil diameter of 4 mm views an extended source of luminance 100 cd m2. Calculate the retinal illuminance in trolands and log trolands. For a particular experiment, 6 log trolands has been designated a safe retinal illuminance when viewing a source. Calculate the maximum permissible source luminance for pupil sizes of 2 mm and 4 mm. A source of luminance 5000 cd m2 is projected by a lens of diameter 4 cm located 4 cm from the source. If the image screen (a Lambertian diffuser) is located 2 m from the projection lens, calculate the reflected luminance from the screen. Compare and contrast visual and physical photometers. Why do spectroradiometers employ mirrors rather than lenses as imaging and beam manipulation components? Why is it more appropriate to use the designations S, M, and L rather than blue, green, and red for the three cone classes found in the human retina? Colors in Thomas Young’s color mixing experiment were additively mixed. Why do printers use color subtraction?