Ординатура / Офтальмология / Английские материалы / Optics of the Human Eye_Atchison, Smith_2000

according to Kerker (1969). This theory applies when the refractive index of the scattering particles is close to that of the medium in which they are imbedded. Applied to spherical scattering particles, this theory predicts that, for spheres, forward-

scatter increases and back-scatter decreases

with increase in size.

The above scattering theories can be readily

applied when the scattering particles are independent of each other. This assumes that

they are weakly scattering and/or randomly ordered. These assumptions do not apply to the cornea, which has an ordered structure.

Cornea

The bulk of the cornea is the corneal stroma (Chapter 2). The stroma contains 200-250 or more layers (lamellae) of long cylindrical collagen fibrils, with the thickness of each

lamella being about 2.0/Jm (Hogan et al., 1971). Fibrils within each lamella have

diameters of 32-36 nm, and are separated by

20-50 nm. The fibrils within a layer are parallel to each other, and uniform in size and spacing (Hogan et al., 1971). The fibrils within a layer are inclined at large angles to fibrils in

adjacent lamellae. The refractive index of

fibres is about 1.47, and the surrounding ground substance has a refractive index of about 1.354 (Maurice, 1969).

Hart and Farrell (1969), using a theoretical model of scattering by arrays of cylindrical structures with the degree of regularity found in real corneas, argued that the transparency of the cornea is due to the regularity of the fibril separation. Their theoretical predictions for the rabbit cornea closely matched experimental measures of transparency. Their theory was based upon the diffraction/ interference from regular arrays. According to

their theory, for an infinite array of equally spaced point-scattering particles, with a separation that is negligible compared to

wavelength, the scattered light destructively interferes in all directions for all wavelengths except in the direction of the incident beam. However, because the fibrils have finite

diameters, their array is not perfectly regular and infinite, and because the spacing is not

negligible compared with the wavelength, there is some residual scattering that is

Passage of light into tile eye 113

wavelength-dependent. It follows that, if this regularity is disrupted further, scattering

increases.

McCally and Farrell (1988) investigated the wavelength dependency of scatter within the cornea and concluded that the range of the ordering of the fibrils (e.g. short distance

versus long distance order) would affect scatter. They argued that if the order is short

range, the dependency is proportional to the inverse of the cube of the wavelength (i.e,

1/,1.3). This was supported experimentally for the rabbit cornea. This cubic dependency is in contrast with the Rayleigh fourth power (i.e. dependency, which assumes no

regularity of the scattering particles and that

the particles have negligible dimensions compared with the wavelength.

Lens

Because the lens is much thicker than the cornea and is composed of cells, we expect it

to scatter more light than the cornea. This happens, with scatter increasing with age, and the forward-scatter being greater than the

back-scatter (Bettelheim and Ali, 1985). Hemenger (1988, 1992) argued that the scattering is caused by the lens fibre lattice.

Fluorescence

Fluorescence is the absorption of radiation by a medium at one wavelength, and the immediate re-emission of radiation at longer wavelengths. The re-emission is isotropic and therefore, in the eye, tends to produce a uniform veiling glare on the retina.

Fluorescence occurs mostly in the lens, and increases with age and in people with cataract

(Lerman and Borkman, 1976; Bleeker et al., 1986; Siik et al., 1993). The lens contains at

least three fluorescent compounds; one called

tryptophan, with a maximum sensitivity at 290 nm, and at least two fluorogens, with maximum sensitivities at 370 nm and 430 nm.

Figure 12.5 shows the spectral emission

properties of these compounds. Tryptophan is present at birth, and does not change greatly with age, but the two fluorogens are not present at birth and accumulate throughout life.

114 Light and tileeye

Wavelength (om)

Figure 12.5. The fluorescent emission profiles of

tryptophan and two fluorogens for a 78-year-old human lens (from Lerman and Berkman, 1978),

The veiling glare produced by fluorescence can lead to a small loss in visual acuity of low contrast targets, an effect that increases with age (Elliott et al., 1993). The level of fluor-

escence depends upon the spectral sensitivity of the fluorescent compound and the spectral profile of the incident radiatiori-i.e. how much ultraviolet radiation and blue light is contained in the illuminating source. The contribution from tryptophan is minimized, because the cornea absorbs much of the activating wavelengths.

Birefringence

Optical materials such as water and glass are isotropic, which means that their physical

properties are the same in all directions. This is because of the lack of atomic or molecular order. In contrast, many crystals have a welldefined atomic order. In some crystals, such as

calcite, the physical properties (in particular the speed of light) depend upon direction of travel. This is referred to as anisotropy,

Materials are called birefringent when the speed of light (or refractive index) depends not only on direction of travel, but also on the orientation of the electric field. This can occur in glass and plastics when they are placed

under stress.

If unpolarized light enters a birefringent material, a phenomenon called double refraction occurs. The incident ray splits into two parts on entering the material, with the

electric vectors of the rays at right angles to each other. These rays are now completely

linearly polarized. One ray is refracted according to Snell's law and is called the ordinary ray. The other ray is called the extra-

ordinary ray, and does not obey this law. Some materials have a particular direction or axis for which double refraction does not

occur, and crystals with this property are called uniaxial crystals. They have two refractive indices, no for the ordinary ray and

lIe for the extra-ordinary ray. The level of birefringence is quantified by the difference

(l1e - 110 ) between these two refractive indices. In biological materials there is usually little crystal-like atomic order, but there is

sometimes order at the tissue structural level. This is called fonn birefringence, to allow

differentiation from the intrinsic birefringence due to molecular arrangement. The eye exhibits some birefringence, mostly due to

form birefringence. This can occur only where there is some regularity to the structure, and it

occurs only in the cornea, lens and retina. In this section, we look at birefringence in the

cornea and lens. Birefringence of the retina is covered in Chapter 14.

Cornea

As mentioned in the above Scatter section, the bulk of the cornea consists of the approximately 200 lamellae of the stroma, in which

the fibrils are cylindrical in shape and are

usually regularly spaced. The fibrils in adjacent layers lie at large and different

angles. Thus, we would expect each layer of fibrils to exhibit form birefringence. According to Bour (1991), Wiener (1912) showed that an assembly of parallel rods immersed in a medium of lower index acts as

a uniaxial crystal, with a positive birefringence with the optical axis in the direction

of the rods. However, because of the large number of layers and their wide-ranging orientation, a light beam incident normally

on the cornea would not encounter fibrils with any overall preferred orientation. Therefore, the cornea would not appear to be

birefringent. On the other hand, a beam inclined to the corneal surface would undergo a slight birefringence. Bour and Lopez Cardozo (1981) estimated corneal birefrin-

gence to be +0.0020.

In contrast with the uniaxial model of the cornea, van Blokland and Verhelst (1987) argued that the cornea acts as a biaxial crystal,

with the faster principal axis lying normal to the cornea and the slower axis lying nasally

downwards.

Lens

The structure of the lens is different from that of the cornea. The lens consists of fibres (about

8 Jim in diameter) that are laid down radially, rather than being parallel as in the cornea.

Therefore, we would not expect the same level of birefringence. Weale (1979) gave values varying between -0.5 x 10-6and -3.5 x 10-6.

Summary of main symbols

n, n' refractive indices on the incident and

refracted side of a surface

R Fresnel reflectance at a surface

T Fresnel transmittance at a surface

References

Beerns, E. M. and van Best, J. A. (1990). Light transmission of the cornea in whole human eyes. Exp. Eye Rcs., 50,

393-5.

Bettelheim, F. A. and Ali, S. (1985). Light scattering of normal human lens III. Relationship between forward

and backward scatter of whole excised lenses. Exp. Eye Res.,41, 1-9.

Bleeker, J. C; van Best, J. A., Vri], L. et at. (1986).

Autofluorescence of the lens in diabetic and healthy subjects by fluorophotometry. Invest. Ophthal. Vis. Sci.,

27,791-4.

Boettner, E. A. and Wolter, J. R. (1962). Transmission of the ocular media. Invest. Ophihal., 1, 776-83.

Bour, L. J. (1991). Polarized light and the eye. In Vision and Visual Dysfunction, vol. 1 of Visual Optics and Instrumentation rw N. Charrnan, ed.), pp. 310-25.

MacMillan.

Bour, L. J. and Lopez Cardozo, N. J. (1981). On the birefringence of the living human eye. Vision Res., 21,

1413-21.

Curcio, J. A. and Petty, C. C. (1951). The near infrared absorption spectrum of liquid water. J. Opt. Soc. Am.,

41,302-4.

Passage oflight into the eye 115

Elliott, D. B.,Yang, K.C. H., Dumbleton, K. and Cullen, A. P. (1993). Ultraviolet-induced lenticular fluorescence: intraocular straylight affecting visual function. Vision Res., 33, 1827-33.

Geeraets, W. J. and Berry, E. R. (1968). Ocular spectral

characteristics as related to hazards from lasers and other light sources. Am. J. Ophthal., 66, 15-20.

Hart, R. W.and Farrell, R. A. (1969). Light scattering in the cornea. J. Opt. Soc. Am., 59, 766-74.

Hemenger, R. P. (1988). Small-angle intraocular light

scatter: a hypothesis concerning its source. J. Opt. Soc. Am. A., 5, 577-82.

Hemenger, R. P.(1992). Sources of intraocular light scatter frominversion of an empirical glare function. Applied Optics,31, 3687-93.

Hogan M. J., Alvarado J. A. and Weddell J. E. (1971).

Histology of the Human Eye. An Atlas and Textbook, p. 89.

W. B.Saunders.

Hulburt, E. O. (1945). Optics of distilled and natural water. J. Opt. Soc. Am., 35, 698-705.

Johnson, C. A., Howard, D. L., Marshall, D. and Shu, H. (1993). A non-invasive video-based method of

measuring lens transmission properties of the human eye. Optom. Vis. ScL,70, 944-55.

Kerker, M. (1969). TheScattering of Light, p. 414. Academic

Ludvigh, E. and McCarthy, E. F. (1938). Absorption of

visible light by the refractive media of the human eye.

Arch. Ophthal, 20, 37-51.

McCally, R. L. and Farrell, R. A. (1988). Interaction of light

and the cornea: light scattering versus transparency. In

The Cornea: Transactions of the World Congress on the Cornea 111. (H. D. Cavanagh, ed.), pp. 165-71. Raven

Press.

Maurice, D. M. (1969). The cornea and sclera. In The Eye, vol. 1, 2nd edn. (H. Davson, ed.), pp. 486-600. Academic Press.

Mellerio, J. (1971). Light absorption and scatter in the human lens. Vision Res.,11, 129-41.

Pitts, D. G. (1978). The ocular effects of ultraviolet radiation. Am. J. Optom. Physiol. Opt., 55,19-35.

Said, F.S. and Weale, R. J. (1959). The variation with age

of the spectral transmissivity of the living human crystalline lens. Gerontologia, 3, 213-31.

Siik, S., Airaksinen, P. J., Tuulonen, A. and Nieminen, H. (1993). Autofluorescence in cataractous human lens and its relationship to light scatter. Acta Ophthal., 71,

388-92.

Smith, R. C. and Baker, K. S. (1981). Optical properties of the clearest natural waters (200--s00 nm). AppliedOpt.,

20,177-84.

van Blokland, G. J. and Verhelst, S. C. (1987). Corneal

116 Lightand theeye

polarization in the living human eye explained with a biaxial model. /. Opt. Soc. Am. A, 4, 82-90.

van den Berg,T. J. T. P., IJspeert, J. K. and de Waard, P. W. T. (1991). Dependence of intraocular straylight on

pigmentation and light transmission through the ocular wall. Vision Res., 31, 1361-7.

van den Berg, T. J. T. P.and Tan, K. E. W. P. (1994). Light

transmittance of the human cornea from 320 to 700 nm for different ages. Vision Res., 34, 1453-6.

Weale, R. A. (1979). Sex, age and the birefringence of the human crystalline lens. Exp. Eye. Res.,29, 449-61.

Weale, R. A. (1986). Real light scatter in the human crystalline lens. Graefe'S Arch. Clin. Exp. Ophthal., 224,

463-6.

Wiener, O. (1912). Allgemaine Satze tiber die

Dielektrizitatskonstanten der Mischkorper, Abh. Sachs. Ges. Akad.toiss. Math.-Phys. Kl., 32, 574(cited by Bour,L.

J. (1991) and Weale, R. A. (1979)).

Introduction

In the previous chapter, we considered the transmittance of the ocular media. Given the spectral transmittance data of the media, we can determine the amount of radiation and

light reaching the retina from a source of known spectral output. The results shown in Figure 12.2 indicate that 50-90 per cent of the

light entering the eye reaches the retina as image-forming light (although it is important to be aware that results from different studies

vary considerably, and there is considerable age-dependence).

Ideally, the spectral and spatial light distribution in the retinal image should be proportional to that in the object. In reality this is not so, because of the effects of aberrations, absorption, diffraction and scatter. Absorption and diffraction are wavelengthdependent, and aberrations, diffraction and scatter affect the spatial distribution. In this chapter, we assess the spatial light distri-

bution in the image.

Retinal illuminance: directly transmitted light

In this section, we present equations that can be used to calculate the retinal light level or illuminance, given the brightness of the object (luminance of large area sources or luminous intensity for effectively point sources). We assume that the eye is correctly focused on

that object, and therefore the equations are not strictly valid if the object is out of focus. If the

level of defocus is small compared with the size of the object, the defocus has little effect on retinal illuminance except near the edge of the object. On the other hand, if the defocus is

large compared to the size of the object, the

equations give very misleading results. This is particularly so for point sources. Campbell (1994) presented a scheme for extended sources that can be used whether the eye is focused or not. Campbell's scheme is for

calculating retinal irradiance, not retinal illuminance, but the equations apply equally to illuminance.

In this section, we consider only directly transmitted light, i.e. image forming light. To determine the retinal illuminance of images formed from this directly transmitted light, it is most convenient to divide this section into two parts; one dealing with on-axis imagery, and the other dealing with off-axis imagery.

On axis

Large area sources

For large area sources, small angle scatter,

aberrations and diffraction can be neglected, and then the light distribution in the image

has the same form as the object. When the eye is observing an object of large angular subtense which has a luminance L (cd/rn-), the corresponding retinal illuminance E' is

118 Li~11t and tilee!le

given by the equation (Smith and Atchison, 1997)

where r is the transmittance of the ocular media ("" 0.6 to 0.9) taking into account light losses due to reflection, absorption and

scatter, n' is the refractive index of the vitreous humour (usually taken as 1.336),and a' is the

half angular subtense of the exit pupil measured at the retina. We can re-express this equation in terms of the (entrance) pupil diameter 0 rather than the angle a' as

where F is the equivalent power of the eye and has a value of about 60 O.

We can also express the retinal illuminance

in terms of area A of the (entrance) pupil, instead of its diameter 0, as

Since the power of the average eye is close to 60 0, and if we take 0 and A in millimetre units, the above two equations can be approximated by the equations

E' "" 0.002830rL02mm or "" 0.003600rLAmm lux (l3.2c)

The troland

The retinal illuminance is sometimes expressed in terms of trolands, which are the

product of the object luminance L measured in cd/m2 and the area A of the pupil measured

in square millimetres. Thus the retinal illuminance E'T in trolands for a source of luminance L and viewed through a pupil of area Amm in square millimetres is give by the equation

E'T= LAmm trolands (13.3a) and since

Amm = rrD2mm/4 then

Relationship between troland and lux

We can relate retinal illuminance E' in lux with retinal illuminance E'T in troland by equation (13.2a) and equation (13.3b). In relating these two equations, we must use the

same units for pupil diameter, and if we

replace the pupil diameter 0 in equation (13.2a) (which is in metres) with the diameter

0mm in millimetres, we have

If we now use the approximation that the

power F is 60 0, we can write these equations as

The point source - diffraction limited

The preceding equations do not apply for very small sources imaged without any aberration. In this case, the geometrical optical image is smaller than the diffraction limited point spread function, which has an angular radius

In general, while the point spread function is

affected by aberrations and diffraction, only the diffraction effect is easily predicted for any optical system.

The total luminous flux in the image of a point source, whether aberrated or not, is

given by the equation (Smith and Atchison, 1997)

where d is the distance of the source and I is the luminous intensity of the source.

For the diffraction limited (i.e. aberrationfree) image, the peak value of the illuminance is given by the equation (Smith and Atchison,

These equations demonstrate two useful results:

1.The total luminous flux in a point spread function is proportional to the square of the pupil diameter (i.e, proportional to pupil area).

2.The peak illuminance is proportional to the fourth power of the pupil diameter or the square of the area.

The point source - aberrated

The light distributions in real eyes and the effect of aberrations and diffraction are discussed in detail in Chapter 18.

Ligl1t leucl at tire retilia 119

Entrance pupil of eye

C\

Light entering eye

Object

Off-axis or peripheral sources

The retinal illuminance of off-axis sources is a little more complicated than for on-axis sources. In conventional optical systems such as a camera, the image plane illuminance decreases as the fourth power of the cosine of the peripheral angle, a result often known as the (OS4 law. This is due to a combination of

the following:

1.The reduction in apparent size of the peripheral pupil (proportional to cos (angle».

2.The increase in distance from the exit pupil to the image on the image plane (the inverse square law, proportional to cos? (angle».

3.The inclination of the image plane to the direction of the incident beam (tilt, which is proportional to cosianglej).

In the case of the eye, the cos" law does not

hold. While the effective pupil area decreases with distance off-axis, approximately as cos(angle), the curved shape of the retina means that the factors 2 and 3 above do not

apply. The curved retina puts the image surface closer to the exit pupil, and tilts the normal to the surface in the direction of the

exit pupil.

How the retinal light level depends upon peripheral angle is important in the measurement of visual fields and the calculation of the risk of the retina being damaged by hazardous radiation sources. Because of this importance, there have been a number of studies in the area (Drasdo and Fowler, 1974; Bedell and Katz, 1982; Kooijman, 1983; Kooijman and Witmer, 1986; Charman, 1989).

Using Figure 13.1, we examine this situation with a simple model of the eye looking at a plane Lambertian source of infinite subtense and of luminance L. Let us consider

Exit pupil of eye

0'

Figure 13.1. Geometry for calculating the retinal illuminance for peripheral large area sources. See text for details.

an annular zone of this source as shown in Figure 13.1. The luminous flux entering the pupil of the eye is given by the equation

where A(O)p is the projected or apparent area of the circular pupil in the direction of the

source. This equation shows that the

luminous flux incident on the pupil depends only upon the solid angle of the annulus, and not on its distance and shape.

If we now assume that this annular light source is focused on the retina, it is imaged as an annular zone on the retina as shown in the figure. The area of this zone is given by the equation

where the internal and external angles are

where m has a value of about 0.82 (Chapter 5). For larger angles, we should use a more

120 Light and the c!li'

accurate analysis (e.g. ray trace through a finite schematic eye) to determine the

relationship, but we will leave this calculation for Chapters 15 and 16.

The retinal illuminance E' is the ratio of flux to illuminated area and thus is given by the equation

In order to calculate values for this equation, we need to relate s', Y' and i' to O. We can do

this as follows. First, from the figure we have

We can find the intersection (Y', 2') of this line with the retina, which we will assume to be circular in cross-section and described by the equation

Figure 13.2 shows estimates of the relative retinal illuminance as a function of off-axis angle for a spherical retina (R' =12 mm) and

the distance a =8 mm for a projected pupil area A(O)p falling off as cos(O). The relative retinal illuminance assuming the cos4 law is

shown for comparison.

Figure 13.2 predicts that, as an object moves off-axis, the retinal illuminance decreases, although not as rapidly as predicted by the cos! law. The simple model neglects a number of important factors:

1.The retina is not spherical.

2.The effective pupil of the eye is slightly larger at peripheral angles than that predicted by the simple cosine reduction (Chapter 3).

3.The exact relationship between the external angle (J and internal angle O~ Equation (13.11) is only accurate for small angles. For larger angles, the aberrations affect the relationship between 0 and 0'.

~0.4

Angle in air (deg)

Figure 13.2. Variation of retinal illuminance with peripheral angle for a large area source, predicted from

paraxial raytracing with a simple model eye.

4.Variation of ocular transmittance with peripheral angle. We expect the transmittance to decrease as peripheral angle increases because of the increase in path length within the lens.

5.Aberrations of the eye increase with increase in peripheral angle, and this will affect the illuminance distribution for small sources.

Kooijman and Witmer (1986)found that the light level in excised eyes reduces more

slowly with increase in peripheral angle than given by the above theoretical predictions

(Figure 13.2). In Chapter 16 we use more realistic models of the eye, with more accurate

estimates of apparent pupil area A( O)p. and of the relationship between external and internal

angles 0 and O~

Retinal illuminance: scattered light

As discussed in the previous chapter, a significant amount of light entering the eye is scattered forward, but out of the imageforming beam. In this section, we examine the light distribution due to scatter from a point source of light. This can be considered in

terms of the retinal illuminance or an equivalent luminance of the image. Because we do

not yet have accurate models of the complex nature of scatter, and because the light level on the retina is too low to be measured

directly, the only determinations that can be made of the scattered light distribution are

subjective measurements.

Several investigators have measured the light level at the retina due to scatter from a glare source in terms of the equivalent veiling luminance. In the presence of a glare source, the threshold luminance of a small patch of light was determined. The glare source was then turned off and the luminance Ly of a uniform background determined, which gave the same threshold for the small patch of light. This equivalent veiling luminance was fitted at the fovea Ly ( 8) by equations of the form

where E is the illuminance in the plane of the eye of the glare source, 8 is the off-axis angle

in degrees of the glare source, and K and n are constants depending upon the particular investigation and the range of values of 8. Setting n to 2, this equation is well known as the Stiles-Holliday relationship. Some specific equations are as follows:

with Ly(8) in cd/m2,E in lux and 8 in degrees. Vos (1984) showed that, in young subjects,

the equivalent veiling luminance is produced in approximately equal proportions by the cornea, lens and fundus. Scatter by the cornea and lens is discussed in Chapter 12, and

scatter by the fundus is discussed in Chapter 14.

Various attempts at modelling the equivalent veiling luminance have been made. For example, Fry (1954)used a reduced schematic

Light level at theretina 121

eye with a refractive index /1' (1.333)and axial length t (20 mm), and derived the theoretical

equation

Lv(8) = (2/3)tR E[cot(8/n') - cos(8/n')]ln'2 (13.17)

where R is a constant. If E is given a value of 1 and R is given a value of 3.39, this equation

gives the same value of Ly(8) at 5° as equations (13.16a) and (13.16b).

Effect of position in the lens of a scattering centre

It is often observed that the effect of a scattering centre is strongly dependent upon its position. For example, posterior polar cataracts have more severe effects on vision than cataracts of similar size that are situated more anteriorly. This has often been explained by the proximity of the scattering centre to the nodal points of the eye, but there is no optical reason why this should affect the scatter properties.

Measurement of angular distribution of scattered light

The light distribution of the image of a point source is known as the point spread function. The shape and width of the point spread

function depend upon the levels of diffraction, aberrations and scatter, and upon the

shape of the pupil. The effects of diffraction and aberration are discussed in Chapter 18,

where it is shown that diffraction and

aberrations produce a point spread function that has a half-width of a few minutes of arc.

On the other hand, scatter causes light to be directed over much wider angles. Therefore,

the light in the periphery of the point spread function is easily identified as scattered light.

Close to the centre of the point spread function, most of the light is from both diffraction and aberrations, and it is difficult to extract the scattered component. For this reason, some investigators of scattered light (e.g. van den Berg, 1995) looked only at that

component of light scattered through an angle greater than 1°.

Various methods have been developed to measure the angular distribution of scattered

light. All of these methods assume that the scattering source has a very small angular subtense, and can therefore be regarded as a point source of light. Two subjective methods are described below.

Conventional threshold method

The essence of this method was given near the start of this section. The threshold luminance of a small patch of light is determined in the presence of a glare source. With the glare source turned off, the luminance of a uniform background is determined that produces the same threshold for the small patch of light.

This luminance is the equivalent veiling luminance, and it is determined for a range of off-axis angles of the glare source.

Flicker method (e.g. van den Berg and I}speert, 1992)

The glare source, which consists of annuli of various angular sizes, flickers at about 8 Hz. One of the annuli is illuminated. If the test patch in the middle of the annuli is very dim, the subject sees the test patch flicker because light is scattered onto the subject's fovea. The subject increases the luminance of the test patch, which flickers in counterphase to the glare source. When no flicker is apparent, the amount of luminance modulation of the test patch equals the amount of light scattered. This is determined for all the annuli of the glare source.

Photon density levels

Sometimes there is a need to express the light

level at the retina in terms of photon or quantal density (e.g. photona/rn'[/s), rather

than in terms of lux or trolands. We can find

an equation for this quantity by proceeding as follows.

Let us suppose that the spectral irradiance at the retina is E' (,1,).1,1, in the bandwidth .1,1,.

The corresponding number N(,1,).1,1, of

photons Zmc/s in the same bandwidth is given by the equation

where h is Planck's constant (6.62620 x 10-34 [.s), c is the speed of light (2.99792 x 108 m/s), and v is the frequency in Hertz. Weighting the

photon density by a weighting function W(,1,) over the entire spectrum, we have

where W(,1,) may be the photopic or scotopic relative luminous efficiency functions, i.e. V(,1,) or V'(,1,). Now from equation (13.2a), for an extended source

where Lr(A.) is the corresponding spectral radiance of the source in WI(st. m3). Equation

(13.19) can now be written in the form

='t1tD2F2 J

Ntot ~ Lr("A.)W("A.)A.dA. photons/rn-Zs (13.21)

Black body source

The form of Lr(A.) in the previous equation depends upon the nature of the source. One type of source of particular interest is the black body source. For a black body source of known temperature T Kelvin, the spectral radiance Lr(A.) from Planck's law is

where

cj =3.7418 X 10-6 W.m2 and c2 =1.4388 X 10-2 m.K.

Real sources

For real sources, the amount of radiation

emitted is always less than that of the black body, and the ratio of the emittances is the

emissivity e. That is